That’s what a bouncer is. And I already used it to solve this problem 20 years ago.
The main point back then was of course that nobody steals your username.
IRC is a message-passing service, not a log synchronization service -- like many of the more modern ones are. You might say that that is one of its architectural strengths, unless, of course, you want to see old messages that were passed when you were offline. In which case you'll need to hack something like bouncers up, which works in practice really well.
IRC doesn't have accounts, you just connect, specify a nickname and it's yours unless it is used by another connected user. So not really possible to track users.
I've been using https://libera.chat/ which allows "claiming" Nicks with a registered email account... I suppose it would be possible to "buffer" messages?
I see some IRCs even keep a mailing-list-like record of all messages posted on a channel... so I don't see how it's not possible.
Logging and nick registration are both extra services that run ontop of IRC.
At one point I extended an IRC server where you could send a list of regex filters. When messages where sent to channels you had joined that matched the filters it would send you push notifications. I think that combined with logging is really all people want on mobile.
Unfortunately someone has to actually run the push server and that
1) Requires more work than volunteers are willing to do
2) Can only be done by people holding the certificates used to publish the app.
I've personally just given up on popular mobile OSes. They're a disaster in nearly every way.
So you'd have to use a server that's connected to the IRC server 24/7 and connect from your phone to that server.