As many others have already said, you are definitely not weird for it. If thats what you enjoy, don’t let others take it from you just because they might consider it weird.

For me personally, I do both. Sometimes I listen to background music while doing other things. As someone said above, its like elevator music. However, for me it improves the mood for some tasks.
On the other hand, I sit down maybe three times a week just for listening to music. I fully follow the emotions and story of a song or album.
The music is quite different between these two modes. For the background music I just need some good tunes. When listening fully, the song must have a text I like or be a really good instrumental song. Then I also need the right song and order of songs depending on mood. Mostly it’s a curated list from which i select song by song. If I hear a great song in background listening I add it to the list. Later I listen to a more songs of the album or artist with full attention and add the ones I enjoy to my list. Funnily there are songs which then have so much meaning for me, I cannot listen to them in background anymore. Either I skip them or i put my full attention to them.