Yes! doesn’t everyone?
I do it. When I sing along with the Beatles, I do a Liverpool accent. When I sing along with bluegrass, I sound like Bill Monroe or Ralph Stanley. (Of course, I’m from a pretty redneck area of Illinois, so that isn’t as big of a stretch as it might seem.) Even in church, when we do the praise song “Shout To The North”, I have a hard time not singing like they sound on the record by Delirious?, which is where I first heard it. They’re British- with a British children’s choir singing back up.
Being from New York, he sounds completely normal to me. 
“Maggie Mae”, on the other hand, is a fun one if your want to give your fake Scouse a try.
Unlike other British Invasion bands, though, the Beatles sound remarkably American in most of their songs, especially the early ones. You can sometimes detect an English accent, but rarely a Northern one.
It’s so common that one online singing teacher recommends singing along with someone to help pick up their style.
I don’t tend to do it when I’m just singing for myself, though. I’ll use the correct style, but not the actual accent.
I simply have to mimic Billy Bragg’s accent when I sing along with him. “I miss the funder, I miss the rine. . .”
And half the fun of singing along to country music is to adopt a pseudo Southern twang.
Hey, that’s when I get to sound like me. 
I “lump together” US Southern accent with Irish/Scottish. So for those, an accent is needed. Since you can’t sing along to a Pogues song without it, the others inherit the trait.