Wireshark saves data. I guess it caches the data and saves it to the same file every few minutes. I doubt it runs as an overlay, but you can look at the data afterwards.
I haven't played any FPS games, just roleplaying games (and many years since that), but I tooled around a few year ago with Everquest: I don't count playing a few hours as "getting back into a game", and tried Guild Wars for a few days. Both gave warning when you were put into a new server.
That being said, I'm not sure you'd get much out of it, at least for some games, because I'm sure many of them, you can only trace to the edge router and not the intranet at the company, so most likely, in that scenario, every server will have the same IP address.. that of the edge router. But I can't say for any particular game if that's the case or not. Plus I think a lot of the "game servers" for games are virtual, and might be running several virtual servers on a single machine for low traffic events, or dedicate several physical machines to a single "virtual game server" if the "virtual game server" is under extreme load.