You recognize the song before you recognize the instrument playing it. Perhaps the original had a full rhythm section and now there is only a guitar. Perhaps a voice you associate with one singer has become a line on a trumpet. Familiarity does the first bit of work, bringing you closer.
After that, try to stop measuring the performance against the recording. A cover needn't reproduce every flourish to take the song seriously. Listen for what has been kept: a rhythmic pattern, the rise of the melody, an awkward turn that gives the tune its personality. Then notice what has had to change.
A smaller arrangement exposes decisions. Without a drummer, the pulse may live in a strumming hand. Without backing singers, a response may become a little instrumental phrase. The performer can leave something out and still preserve the part that matters to you.
There is a social advantage to recognition, too. People who would never choose the same album may share one remembered chorus. That common ground can make an audience out of a group of strangers. It doesn't entitle anyone to demand that the rest of the set follow their own memories.
Stay for the next song if you can. The familiar cover has opened a door, but the musician may have something less recognizable on the other side. Hearing that piece gives you a chance to encounter their taste, not only your own.
