Spontaneous songs pop in my head and out of my mouth all the time with no warning. I feel ya. I used to think I was picking up radio signals in my dental work, but now all that metal is gone and it still happens. It’s rarely a song I like, but almost always a song I know every word and I can’t shake it until I go through the entirety of the song including the bridges, choruses and solos.
playing entire songs in your “head” seems to me pretty easy, almost automatic like you describe. Your brain is like a tape recorder so if people listen to music a lot it stands to reason they should be able to recall music/lyrics they prefer in great detail.