Hi EnquiringMindHi taraa, no, you can't use "abandon" in that context. It's difficult to explain, but maybe it's that "give up" in this sense means "relinquish" or "surrender". It implies a sort of ethical or altruistic or morally positive act. You do something for the benefit of someone else, so that someone else benefits from your gesture - here the woman benefits. "Abandon" doesn't carry any notion of altruism or doing something for someone else's benefit. You do it for your own benefit only.
Thank you so muchGoogle gives ONE actual result for this, so on one occasion someone used the word wrongly. It happens. If it was possible to use the word "abandon" in that sense in that collocation, you'd expect to see it more than once.
Thank you so muchYes. By giving up his seat he is clearly voluntarily doing a favour to someone else, acting altruistically. That's not the sense of "abandon". As I said, if "abandon my/your/his/her seat" (in the sense of voluntarily giving it up as an altruistic act) was normal usage, you'd expect to find more than one example on Google.