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When trying to understand what dark matter is, it is helpful to know that some properties of it can already be derived from various observations, such as, it only interacting via gravity and no other force, it being five times more in amount than regular matter, etc. But one thing that has confused me is the presumption that it must have mass.

Dark matter having mass seems to be derived from the fact it can interact only via gravity, but considering that the mass-energy equivalence exists, this means that in the context of General Relativity, where gravity is described as the curving of spacetime, it can also apply to massless, energetic particles, not just massive objects. So couldn't dark matter very well be massless?

Quantum Wonder
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Dark matter must have mass because it is defined that way. To quote Wiki:

In standard cosmological calculations, matter means any constituent of the universe whose energy density scales with the inverse cube of the scale factor, i.e., $\rho \propto a^{-3}$. This is in contrast to radiation, which scales as the inverse fourth power of the scale factor $\rho \propto a^{-4}$, and a cosmological constant, which does not change with respect to $a$ ($\rho \propto a^{0}$)

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In principle, "dark matter" means all components of the universe which are not visible but still obey $\rho \propto a^{-3}$. In practice, the term "dark matter" is often used to mean only the non-baryonic component of dark matter, i.e., excluding "missing baryons". Context will usually indicate which meaning is intended.

If dark matter were massless, then it would move at the speed of light, and then it becomes radiation (i.e., it would obey $\rho \propto a^{-4}$). It would no longer be dark matter, but "dark radiation".

As you might expect, dark radiation behaves differently from dark matter, and would leave different imprints in cosmological data. That's why we are able to say that, as of right now, there's very little dark radiation in the universe; certainly much less than dark matter.

Allure
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