Learn English – Does “more than” mean “over”

This is a very interesting question. When politicians say:

The rich need to pay their fair share [in taxes].

they are referring to some amount of tax they feel would be equitable, and stating that the rich should be paying that amount. (In all likelihood, they also mean – at least in their opinion – the rich aren't paying that amount yet, but they should be.)

However, when talking about doing a fair share of some activity:

Greg has done his fair share of traveling over the past few months.

the meaning can be somewhat idiomatic. That sentence doesn't necessarily mean that Greg has done more traveling than he should have, or that maybe he should have given up his airline seat to Deborah, who hasn't been away from home in years. It simply means Greg has done a LOT of traveling. One paraphrase would be:

Greg has sure traveled a lot over the past few months!

In this usage, fair doesn't necessarily mean "what is right", or "what is acceptable", or "what is within standards," or "what is equitable" – though it could, depending on the context. For example, consider these two conversations; the first is between Greg's neighbors, who are talking at the mailbox:

Ms. A: I hear Greg is going to Italy next month.
Mr. B: Really? Didn't he just get back from Ireland a few weeks ago? Greg has done his fair share of traveling over the past few months.

Chances are, fair share is being used here idiomatically – it's not like the neighbors feel as though Greg is doing an unfair amount of traveling. But watch what happens when we move the conversation to the conference room at Greg's office:

Manager: We need someone to go to Italy next month.
Greg [raising his hand enthusiastically]: I'll go!
Deborah [more meekly]: I'd like to go...
Greg [cutting Deborah off]: No, that's okay, I'll take this one.
Assistant Manager [sternly]: Look, Greg, didn't you just get back from Ireland? And weren't you in New Zealand the month before that? And we sent you on that trip to Ecuador back in June, too. [to the manager] I think Greg has done his fair share of traveling over the past few months. We should send Deborah on this trip.

If you notice, the assistant manager has uttered the same sentence as Mr. B – word for word – but the meaning of fair has shifted. This time, the assistant is using the word to mean "what is equitable, or "what is more even."

When you say:

I broke up with her and she cried over me.

it means she was crying because you broke up. It's your fault.

The object of cried over is generally the reason why.

I broke up with her and she cried all over me.

To me (AmE) this does not sound natural, but pretty much it is literal. She was crying and the tears fell on you. But we don't know why she was crying.