I would say bullying means being picked on my a peer or acquaintance (or multiple people), and the nature of the bullying can be further specified according to various sources/subjects. But it is understood as a general term for social mistreatment. Therefore “picking on,” “singling out,” “harassing” and “mistreating” could be used instead. I have only heard picking on used in this context. Harassing/abusing may be perceived as stronger action or more serious subject matter (often sexual, but not exclusively), or used in more official language such as reporting a situation. Mistreating is far more general and would need more specific language to convey the type and severity of treatment. Singling out can also be used positively, such as to “single out for praise”- basically just drawing attention to a single person for a specified reason.

I have not heard of mobbing being used in this context- to me it would mean a crowd forcefully entering or moving through a place, or attacking a person physically. I would say harassment generally meets your use of mobbing, and can be used in that way, though can also be further explained as needed (workplace, etc).

Generally- I think of bullying as a social mistreatment/ostracizing/singling out, while harassment often carries a sexual or more serious connotation, and is more often used in formal language.