It used to be that on a Mac, you could set up multiple iChat accounts (since it was actually run on the AOL AIM backend), and login/out of as many as you wanted simultaneously, and in that case, you could’ve created two conversations with the same person ( I used to do exactly this for ‘business’ and ‘personal’ purposes; especially where preserving communications was mandated by corporate guidance or actual laws).
AIM, of course, was killed several months ago; so all my nifty, handy AIM handles will no longer login (but thankfully the chat history is still there).
To the best of my knowledge, there is no longer any way to use to separate iCloud/AppleID accounts with Messages; only one can be logged into iCloud on the Mac user account at any given time.
Under Sierra and High Sierra, you can still add a Jabber account (https://www.jabber.org/faq.html), but that support has been removed in Mojave, so if you plan to go there, it’s a dead end.
Additionally, Messages on iOS only supports the one iCloud account, but you didn’t specify Mac only, so I don’t know what to say there.
Basically, to do what you’re asking about, you’d need to add in a second chat service, like WhatsApp (encrypted, free), or similar (maybe Google Hangouts, if you don’t have privacy concerns); and either setup multiple accounts under the same app (WhatsApp supports multiple logins), or use one app for one conversation, and one for the existing one.
‘Delete Conversation’ will do exactly that: it will delete not only the contents of the chat history, but remove the entry for that conversation from the sidebar. That person’s name/avatar will not reappear until you start a new conversation.
‘Clear Char Transcript’ will clear the entire chat history, but will leave the person/chat window in the sidebar.
To the best of my knowledge, no, however, Apple has made HUGE changes to iChat cum Messages cum Messages in iCloud, so I wouldn’t want to make any bets on that. In Sierra (High Sierra?) and iOS 11, Messages all allegedly moved to Messages on iCloud, and that is “where the truth is” now. There are still local transcripts on both Mac and iOS, but they are supposed to merge into iCloud so that “the truth” is available on all devices; however, that is asbsolutely still not the case.
I have several contacts with whom conversations occuring on my iOS devices don’t necessarily every appear in the history on my Mac (which upsets me greatly); but usually conversations on my Mac do merge and appear on iOS. I have the most problems with (surprise) Android/standard SMS users (green bubbles); but it does still happen with fellow iOS users (blue bubbles).
Don’t even get me started on group chats.
Anyway, if you are for some reason driven to clear your current chat for, I don’t know, privacy concerns about past sensitive issues discussed, but also don’t want to forever lose the conversation history for your own possible needs in the future, you can save the transcript out separately, and perhaps even put it on a password-protected disk image, or make a password-protected (Locked) Note in notes.app, so that it syncs and is available on all your devices, and exists (securely encrypted) on iCloud, in case you ever lose all your devices.
Once you’ve saved the transcript history (just in case) you can ‘Clear Transcript’ or ‘Delete Chat’; they will both have the same effect.
To save the entire conversation (from a Mac), scroll to the top of the conversation, then keep re-scrolling up and up and up (sadly, Page Up and Home buttons do not work; you will have to scroll by hand using the scroll bar or touchpad) until Messages has finally loaded the entire history (this can take a very long time if you are both verbose and have a long history in time); once you have completed this odious task, just choose print (Command-P); then ‘Save as PDF’.
– If your history is super big, and filled with images, be prepared for a crash when it tries to create the PDF.
– Other file attachments will not be saved; if they are important, locate them first (they should originally have landed in your Downloads folder).
– external links to websites may not function, especially if Messages loaded the helpful preview; review any critical links before deleting.
– I’m not sure what will happen with GIFs, movies, video, Memojis, Animojis, but I suspect you will at best get a preview thumbnail only; however, in Messages.app you can right-click on any such object and Add to Photos as desired before you delete the chat.
– again, there may be part of your chat history that never merged from other devices, depending on whatever it depends on; e.g., your pal might at some point have used Android/SMS instead of Messages/iChat, and that portion might not have merged to your Mac, so, I dunno, have fun comparing the history on your iOS devices first/also?
This can continue deep into technical weeds; if you need further advice, you’ll need to be more explicit as to your goals.