Rob forcibly grabs an item from Bob's hands, genuinely believing that it is his (Rob's) item and that Bob is possessing it unlawfully. Rob does so while knowing that Bob contends that the item is his (Bob's); however, Rob is sure that Bob is wrong (either mistakenly or consciously).
Does Rob commit a crime e.g. robbery? Does it matter who the actual owner (or lawful possessor) of the item is?
The question you pose relates to the theft component of robbery. If the taker believed the item was theirs, then the mens rea for theft was not present, and therefore, there was also no robbery.
Every one commits robbery who
(a) steals, and for the purpose of extorting whatever is stolen or to prevent or overcome resistance to the stealing, uses violence or threats of violence to a person or property;
(b) steals from any person and, at the time he steals or immediately before or immediately thereafter, wounds, beats, strikes or uses any personal violence to that person;
(c) assaults any person with intent to steal from him; or
(d) steals from any person while armed with an offensive weapon or imitation thereof.
To "steal" is defined to mean "to commit theft."
Theft is defined at s. 322:
(1) Every one commits theft who fraudulently and without colour of right takes, or fraudulently and without colour of right converts to his use or to the use of another person, anything, whether animate or inanimate, with intent ... (a) to deprive, temporarily or absolutely, the owner of it, or a person who has a special property or interest in it, of the thing or of his property or interest in it ...
...
(4) For the purposes of this Act, the question whether anything that is converted is taken for the purpose of conversion, or whether it is, at the time it is converted, in the lawful possession of the person who converts it is not material.
The hypothetical you raise goes to the mens rea of the underlying theft, not anything special relating to the heightened offence of robbery.
The mens rea for theft is (R v Rowan, 2023 ABKB 164, para. 46; citing R. v. Lafrance, [1975] 2 S.C.R. 201):
If the taker believed the item was theirs, then the mens rea for theft is not present (R. v. Senior, 2021 ONSC 2729, para. 269):
The taking or converting is not done fraudulently and without colour of right if the person taking it did not know that he was not legally entitled to take or convert the property. Even if mistaken as to his entitlement, the person must be found not guilty.
Depending on the manner in which the item is taken, the act could still constitute another offence (e.g. plain assault) even if not theft or robbery. And as always, the actual owner would still have a civil remedy to have the item or its value returned.
The German criminal code defines:
§249 (1) Wer mit Gewalt gegen eine Person oder unter Anwendung von Drohungen mit gegenwärtiger Gefahr für Leib oder Leben eine fremde bewegliche Sache einem anderen in der Absicht wegnimmt, die Sache sich oder einem Dritten rechtswidrig zuzueignen, wird mit Freiheitsstrafe nicht unter einem Jahr bestraft.
(shortened translation: Taking away someone else's property with the intention of keeping it for yourself or another)
So yes, this would qualify as a robbery. However, §16 deals with mistakes of this kind ("Irrtum" in German) and §15 defines that only intentional actions are punishable (unless explicitly specified otherwise). §249 doesn't do so.
So, in summary: You would not be punished according to German law, though legally speaking it does qualify as a robbery.
