If I choose to hang out in chat.cooking and chat.unix at the same time, I'm going to be confusing at least one group as to who I am.
If you're aiming for least confusing, then pick a name and stick with it. Otherwise, accept that you'll probably end up confusing someone.
Right now, SE chat is a mostly-separate system from the individual SE sites, with some periodic synchronization. You have a separate profile on Chat that is associated with one of your SE site accounts, from which it pulls your name, picture, etc. Users who've signed into chat and have chat accounts can be invited into rooms, notified of comments, etc.
Chat rooms also have an associated SE site, from which they take their color scheme, tags, etc. By default, visiting Chat from the link on a SE site will present you with a list of all rooms associated with that site. Note that rooms can and do exist which are not associated with any SE site.
So this suggestion could be implemented in a number of ways...
Room association overrides user association: Upon entering a room, the system checks your account for a possible connection to the site associated with the room. If you have an account set up on that site, your name and picture from that account are used in favor of those associated with your chat account.
Upside: Chat remains separate, multiple names, images just work.
Downside: Potentially confusing for users who click through to your profile and find little connection to the user they were just chatting with. Impossible to search for your per-site name in the Chat user directory.
Per-site chat accounts: Like how Stack Overflow and Meta Stack Overflow have (but perhaps without the completely separate systems). Visit chat.sitename and you're signed into a matching account, visible in the user directory, and with no confusion about who you are when you're in a room vs. elsewhere. Only rooms for the site you originated on would be available (to enter other rooms, you would need to enter chat from a different URL).
Upside: Very straightforward behavior for users who don't use a lot of different sites. Closes "20 rep points on one site lets you chat on any site" loophole.
Downside: Can't chat with (or even find) users from other sites unless they visit your site first (or you visit theirs). Site moderators would need to be more active in chat, since there would be no cross-site chat moderation (not necessarily a down-side).
No chat accounts: Like "per-site chat accounts", but with no separate account system at all. You are who you are on the site associated with a room; if you don't have an account there, you don't chat. The user directory is the one on the site.
Upside: fewer accounts to worry about. Ability to ping / invite anyone on the site, even if they hate chat.
Downside: No inviting users without an account on your site to chat with you. If you're on the site, you're available to chat. Huge and probably entirely unrealistic change to how Chat works. Bears.
Personally, I don't see a big advantage to #1, and #2,3 sound like a ton of work for somewhat dubious advantages. The separate Chat.SO thing already confuses people.