Still have not managed to fix the issue. Patched it for now by turning off 2FA through the LastPass app and enabling the google authenticator. Only thing that is missing is the push notifications, but to be honest I am happy that it just works at this point...
I had the same issue...all my Authenticator accounts disappeared, and clicking on Recover Account took me to a prompt in Lastpass asking to select which account to use (even though I was already logged in!).
I deleted both apps from my phone, downloaded and logged back into Lastpass (using SMS authentication), then opened Authenticator and was able to recover everything.
Clearly a bug, but if you're able to use an alternative form of authentication to reinstall the Lastpass iOS app, this fix worked for me.
I have had the Lastpass authenticator app on IOS lose its account settings multiple timees and across two iPhone models now. It might be an IOS 15.0 thing but it has happened to me so many times now I cannot remember if it happaned in IOS 14.0. I can fix the problem by using the recover account into link in the app. It then pushes you over to Lastpass bu the only way to make things work for me is logging out of Lastpass and then logging in and using SMS as the second factor. Ideally you would want to disaable SMS factor as it is not as safe but one dare not with this problem. I am guessing that for some users something like that doesn't even work. I too have been a long term Lastpass user since approximately 2008. Since the LogMeIn acquisition it has been price hikes and minimal attention to development, it is all but a cash-cow for LogMeIn. I have thought about BitWarden as an alternative but the good 'ol user lock-in comes into play here and the sheer hassle of setting up an alternative. The Apple watch push notication is super helpful but I they cannot fix authenticator and stop if losing its account info, I would think a switch is in order because for me that and the lockin is about the only thing keeping me with Lastpass these days. There are now a number of credible alternatives.