Common law and civil law treatment of women's testimony
Neither common law countries nor civil law countries make any formal distinction between the testimony of men and the testimony of women. There may have been some point in the distant past (see also here) where this wasn't the case, but equal treatment and men and women's testimony, at least as a formal rule of evidence matter, has been the case for hundreds of years in both of the main Western legal traditions. My best estimate, from several notable examples (the trial of Joan of Arc in the 1430s CE is another), is that the categorical exclusion or devaluation of women's testimony in Europe ended sometime around six hundred years ago, in the early modern period, as Europe emerged from the Middle Ages.
Both common law and civil law legal systems mostly derive from pre-Christian Roman and, in the case of the common law, also Germanic legal concepts, even though Europe was predominantly Christian when these legal traditions came into being, and the religious texts of the Jews and Christians do not espouse gender equality in the modern sense (and neither did pagan Romans or Germanic tribes).
In part, the formal equality of men and women's testimony, despite this fact, is a function of the both traditions' tendency to firmly place credibility determinations in the hands of finders of fact (juries in common law jury trials, judges in common law bench trials and judges in all civil law trials), on a case by case basis, rather than according to categorical rules of evidence. So, as women have come to be seen as more equal to men, their credibility in testimony has increased on a case by case basis over time, more incrementally and less formally.
There are spousal immunities (which date to the late 16th century) against testifying against your spouse in some cases, but those are currently gender neutral and have been for a long time (at least since sometime in the 19th century with the demise of the doctrine of coverture).
Islamic law treatment of women's testimony
The reason for the distinction between men's testimony and women's testimony in Islamic law arises from authoritative Islamic religious texts that say so, in a particular context. But this is only partially for any articulated legal or logical reason, and in the view of leading Islamic law scholars it exists primarily because, God, communicating through God's earthly representative, the Prophet Muhammad, said so.
This is confirmed in the answer from Behzad Rajaei, who attributes one of the main distinctions between the weight of men's testimony and women's testimony in some areas of Islamic law as flowing from verse 282 of Surah Al-Baqara (a surah is one of the 114 "chapters" of the Qur'an itself, within which there are units commonly translated as "verses", as opposed to a hadith which is not part of the Qur'an but is part of the larger Islamic tradition), and noting that this standard is ta'abbudi (i.e. it is derived from a textual provision (The Qur’an and hadith) in the that must be obeyed even though the person obeying it does not know the rational reasons for it). This is in contrast to parts of Islamic law that are called ta'aqquli which are derived from rational reasons. Wikipedia recounts an additional hadith (Sahih Bukhari: Book of Witnesses: Chapter witness of women: Hadith no. 2658), and commentary from Islamic legal scholars regarding this issue.
The linked verse is presented at the link in both the original classical Arabic, which is authoritative, and an English translation. But the English translation is not reproduced here, so as not to offend the subset of Muslims who take that position that it is improper to use translated versions of Surahs.
In substance, verse 282 of Surah Al-Baqara pertains to the process for having a scribe document debts at the direction of a debtor, which should be witnessed either by two men, or if two men aren't available, by one man and two women so if one of the women forgets the other may remind her.
At the time that of the Qu'ran (i.e. in the 7th century CE), most men and more women in Arabia were not able to read and write, and illiterate people relied upon scribes to commit agreements to writing with witnesses who may have themselves been completely or mostly illiterate. (Literacy was also not widespread among common people who were not clergy in most of Europe in the 600s CE, well after the fall of the Western Roman Empire.)
A commentary to this critical part of the linked verse (which since it is not the text itself, is appropriate to quote) states:
Generally speaking, there is a difference between witnessing and
giving testimony before a judge. Verse 2:282 talks about witnessing a
debt contract, not giving testimony. To fully understand the context
of this verse, we need to keep in mind that 1500 years ago women did
not normally participate in business transactions or travel with
trading caravans and, therefore, not every woman had the expertise to
witness a debt contract. Even if two women were available at the time
of signing the contract, perhaps the primary witness might not be able
to recall the details of the contract or appear before a judge because
of compelling circumstances such as pregnancy or delivery. In any of
these cases, the second woman will be a back-up. Some scholars
maintain that one woman can be sufficient as a witness so long as she
is reliable. As for giving testimony, a ruling can be made based on
available testimony, regardless of the number or gender of the
witnesses. For example, the beginning of Ramaḍân is usually confirmed
by the sighting of the new moon, regardless of the gender of the
person who sights the moon. Also the highest form of witness in Islam
is for someone to testify they heard a narration (or ḥadîth) from the
Prophet (ﷺ). An authentic ḥadîth is accepted by all Muslims regardless
of the gender of the narrator. Moreover, if a husband accuses his wife
of adultery and he has no witnesses, each spouse must testify five
times that they are telling the truth and the other side is lying.
Both testimonies are equal (see 24:6-10). In some cases, only women’s
testimony is accepted while men’s testimony is rejected, such as
testifying regarding a woman’s pregnancy or virginity.
Also, while Behzad Rajaei correctly stated the majority view that this requirement is ta'abbudi, the fact that the verse itself contains the reasoning behind the distinction, which is further elaborated upon by the commentary above, there is some room for a minority interpretation that this verse is actually ta'aqquli and could be deviated from in cases where the articulated reasoning behind it, viewed in its historical context, does not apply.
I don't know, however, if any of the leading Sunni schools of Islamic law (i.e. Hanafi, Maliki, Shafi'i and Hanbali), or Shi'ite interpretations of Islamic law taken by any Shi'ite sects (e.g. Twelver a.k.a. Jaʽfari jurisprudence, Ismāʿīlī, or Zaydī), actually take this position. But it is the position taken by Muslim modernists who follow the Egyptian reformer Muhammad Abduh in viewing the relevant scriptural passages as conditioned on the different gender roles and life experiences that prevailed at the time rather than women's innately inferior mental capacities, making the rule not generally applicable in all times and places. See Mohammad Fadel, "Two Women, One Man: Knowledge, Power, and Gender in Medieval Sunni Legal Thought" 29(2) International Journal of Middle East Studies 187 (1997).
There are four major schools of Sunni Islamic law, and multiple sects within Shiite Islamic law, and there are many points upon which they don't agree. Quite a few of these disagreements involve the roles of women. Some schools of Islamic law see some aspects of how women are treated in some predominantly Islamic countries (such as honor killings, and specific articulations of what the modesty in dress required in authoritative Islamic law sources means in practice) as a matter of local pre-Islamic regional cultures, rather than as a religious legal mandate. But it is fair to say that equal and identical treatment of men and women is not a foundational principle of Islamic law in any of the major interpretations of Islamic law, and that all of them involve patriarchy to some degree. One way that this distinction is expressed is that:
Gender roles in Islam are simultaneously colored by two Quranic
precepts: (i) spiritual equality between women and men; (ii) the idea
that women are meant to exemplify femininity and men masculinity, but
that neither is superior in the eyes of Allah except in moral
righteousness and actions.
The specific rules that distinguish between men and women, however, in the legal sphere, are perceived by Westerners as favoring men.
The Usuli school of Ja'Fari jurispurdence, which is the most widely followed Shia school of Islamic law and most relevant to Iranian Islamic law jurisprudence as a result (as the Republic of Iran's theocracy is a Shi'ite Islamic theocracy), is noted for making "legal interpretations were kept flexible to take account of changing conditions and the dynamics of the times" (according to the The Oxford Concise Dictionary of Politics (2003) at page 487), so it might be more open to this idea than many other schools of Islamic legal thought. But given that the 11th century CE Islamic law scholar al-Shaykh al-Ṭūsī, cited by Behzad Rajaei in his answer is a leading figure in this school of Islamic legal interpretation and takes the opposite position, there is probably no major school of Islamic legal interpretation, and in particular, is probably no school of Islamic legal interpretation that is relevant in Iran, that does so.
From a secular historical perspective, the treatment of women's testimony in Islamic law may represent one of several cases (the Islamic law stance on polygamy, where men were limited to four wives and discouraged from having more than one due to the difficulty a man would face treating them equally in Islam, while no such restriction were present in the preceding pagan Arabic customary law, is another), where Islamic law, by considering women's testimony at all, treated women better than the immediately preceding pagan Arabic customary law did, but not equally either. While this is merely a speculative observation, categorical rules may have been a mechanism for these half-reforms, because if informal credibility determinations on a case by case basis had been relied upon, the pre-Islamic prejudices of pagan Arabic customary law would have thwarted any reforms at all. But these half-reforms were frozen in time, rather than evolving further, by the early documentation of Islamic law's main sources that were faithfully preserved over time (even though the dictates of the Islamic prophet were originally all delivered orally, according to Islamic tradition).
Islamic legal scholarship has not (yet) managed to find ways to mitigate the divergence between historical religious texts and modern norms with interpretive glosses to the same degree the religious law scholars in Rabbinic Judaism have in their Talmudic tradition.
The widespread perception in the West is that Islam devalues women and treats women horribly, in general, and that unequal treatment of their testimony is just one of many symptoms of that. Indeed, this is perhaps the foremost criticism of Islam in the West.
Most Muslims, and most mainstream Islamic law scholars, would say otherwise, even if they themselves favor some reforms in how women are treated in their communities. One would think that an Iranian law attorney would be better informed than any regular contributor at Law.SE regarding how this is viewed within Islamic law and the overall Islamic religious-legal worldview.
