This is a completely different topic to password management, IMO, but Firefox has a great feature to handle this situation - container tabs.
Basically, each tab can run in its own isolated environment, which means you can be logged into, for example, different Google accounts on the same browser at the same time. I use it for work, as I mostly use Google docs for work, but never YouTube, Gmail, etc. So I have it setup that whenever I go sheets.google.com, my "Work" container opens and I'm already logged into my work Google account in the new tab. If I go to youtube.com, it automatically opens in my "Personal" container and I'm logged into my Gmail account.
You can also override this on an ad-hoc basis, by right clicking the link or bookmark, and selecting which container you want to open it in. So if I ever did want to view Youtube on my work account, that's easy to do without messing up my container automation.
Another benefit of this is you can have Facebook in its own container, and
only login on the Facebook container. That means you can click links and they open with you already logged in. But if you're on another other site, you're not logged in, so FB can't track you with it's Like button and embedded pixels.
It's a great feature, and not that well known.
For you
Epona, password managers can easily have more than one login, sure. I can't remember how LastPass works, but for Bitwarden, it's does a lookup of the URL and compares that with your password vault. If there's a match, it offers to log you in. If there's multiple, you just choose the one you want at that time.
I just checked and I actually have 7 Google accounts. One personal, and various work ones, and some for the clubnight I run. When I go to Google, if I wasn't logged in, I just click the Bitwarden button, and the 7 logins show up. I click the one I want and it auto-fills the username and password for me, just like any browser autofiller.
For multiple browsers with different logins for each you could handle that by renaming the login. Here's my urban login, for example:
View attachment 409035
So if I wanted to login using my firky troll account when on Chrome, but stay as fez909 on Firefox, I can add a second login and change the "Name" bit so I know which one to use. Now, you can manage your cross thread beef really easily without detection. Just go to urban, open Bitwarden and you'll see all your accounts. Select the one that matches your persona and browser for that day, and continue your mission to piss everyone off.
View attachment 409036
So, it's not automatic...but it's easy.
For games etc, it's not quite as handy. There are plugins for all the major browsers to handle web stuff, so that's no problem. And I have an app for Android that detects when I enter a password field on almost every app, and then I get a popup asking if I want to use Bitwarden to fill it. It takes me out of the app, but it's pretty painless. Then I copy the password from the Bitwarden app, press back and paste into the app I'm trying to login to.
For desktop apps, it's not as good. You will have to switch to the browser extension or website, search the vault, click the copy password button and switch back to your app. It's definitely the worst thing about password managers, but it's actually not a big deal. It was this that stopped me using them for years, and then when I finally did switch, I found it wasn't that much of a problem.
I think you'd definitely benefit from container tabs on Firefox, assuming FF can run your games (I find it's slow for some JS stuff). Everyone would benefit from a password manager