Yes, it's like reading for me, in that I'm "fluent" - the meaning of the music just goes straight in and I'm not conscious of the decoding/blending process (to use a phonics analogy). There's no real conscious thought there - it really is a natural as reading for me. Likewise, I often find myself scanning a piece of music (as you would a page of text) or, as Whattodo said, reading a bar ahead of what I'm actually playing. This is actually a vital skill for a musician, anyway - you need to know what's coming and plan for things like dynamic changes, page turns, etc. So I guess I sort of have three things going on in my head at once - the music I'm actually playing, what I'm reading ahead to and thinking about, and then anything to do with the performance (phrasing, tuning, alternative fingerings, listening to other people in an ensemble, etc).
It sounds complicated, but I guess it's not any more so than driving a car - as a beginner, you're thinking "mirror-signal-manoeuvre ... the engine sounds a bit strained, I'd better change up ... clutch down, shift, oops that's fifth, not third..." all the time, but after a while you handle the car more instinctively and can devote more brain space to looking and planning ahead on the road.
I learned to read music very young, so I can't remember not being able to read it (just as I can't remember not being able to read print). So for me, it's almost like another language. When I was a bit older, did have to spend some time learning the "grammar" properly (hello grade 5 theory!) but that's separate from fluency in reading, for me. It is something you get better at with practice, and you can also lose skills if you don't practise them: I did A-level music, so at one point I was good at score reading (where you have to read and play four lines of music at once, often in different clefs). I'm pretty rusty at that now! But I don't think I'll ever lose my baseline ability to read one or two lines of music fluently (short of getting dementia
).