Lemmy and Mastodon servers are capable of speaking to each other using the ActivityPub protocol, but they present substantially different APIs and user interfaces to the end user. There is no way to use a Lemmy app to log in to a Mastodon instance, or vice versa.
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They have a completely different UI, so I don't think that's likely, but I might be wrong.
Accounts are local to the instance they are created on, so unless Lemmy provides an API the mastodon client can talk to (which AFAIK it doesn't) you can't use one client with a different server. Posts and other things (likes, videos, images, comments, etc) are federated between instances so you will be able to see posts. How the different apps (Lemmy, Mastodon, Pixelfed, Peertube, etc) interpret and choose to display those, as well as how they interact with those (e.g. allowing you to comment on a comment vs only on a post) vary across the different offerings.