The phrase share of, particularly when preceeded by a pronoun, refers to some amount that is rightfully owed to someone, or expected for someone, like when a pirate talks about his share of the treasure, or a member of a team of bank robbers talks about his share of the cash. However, the expression is often used metaphorically to refer to some unspecified amount. For example, a manager of a failed project might talk about his share of the blame.
In this case, my share of women means that the character has been lucky in love. He hasn't just had two or three girlfriends, he's had several. And when he emphasizes that by humorously adding, "In fact, I've been with like a lot of people's share of women," he means that, whatever number someone's "share of women" would be, you could probably triple that, and that's how many women the character has been with. In other words, he's a self-described Casanova.
So, the expression "I've been with like a lot of people's share of women" is intended to be a humorous way of saying that he has been in a lot of relationships with women, probably mostly short-term relationships. Yet, in spite of all this "experience," he's never felt a lasting, close, affection like Ross has for Rachel.
I've never watched Friends, but I'm guessing that these lines are meant to be both funny and touching at the same time.