LastPass Authenticator (I am NOT discussing LastPass the password manager!) is an implementation of a Time-based One-Time Password (TOTP)
generator. It starts with a random seed that is set by scanning the QR (bar) code during the initial setup. After that, it computes a 6-digit code every 30 seconds based on the initial seed and the current time. When you log in to a service that uses TOTP as its second factor, the service knows the current time and the initial seed, and can run the same calculation as your app. The "authenticator" app does this calculation entirely on its own; all it needs to know is the (approximate) correct time. You can think of it as sharing a big list of "random" codes, with one crossed off every 30 seconds. The idea is that someone who doesn't have your list (app with seed) can't figure out what the current code is.
Google Authenticator made this popular, but there are several alternative apps that all use the same scheme and understand the same QR codes.
Some "authenticator" apps add value in other ways. LastPass Authenticator provides backup/restore and also, with some sites, it can send a request to your phone and the app simply asks you to confirm, rather than making you type in a code. Duo Security uses this as its primary method, though it can also do TOTP codes. I'll admit it has been a while since I tried LastPass Authenticator, but I never could get the auto-request thing to work.
What I like about Authy is that, not only can one do backup/restore, but it can also sync the seeds across different trusted devices. So if I am on my desktop (as I am now), I can open the Authy application there to copy a code that I paste into a web browser. (You don't have to use this feature, but I find it convenient.) I see that LastPass Authenticator offers a Windows Store app, so that's similar.
Again, this has NOTHING to do with the LastPass password manager other than that it supports TOTP as one of a variety of multifactor authentication options.
Back to your original question - none of the "authenticator" apps I know of provide a way of exporting or importing the seeds from a different app. If you switch apps, you'll have to disable 2FA on each site, then re-enable it with your new app.