I had only movable Do for decades. Some primary teacher taught it and that's all I ever learned. Ajs, Ti Do, when you learn it that way, is not the same as B,C or F#,G (in G major). The distance between Ti Do and Mi Fa is closer than a semitone. It makes you feel the movement more, especially in the absence of harmony. String players use it to. It's one of the tunings. I didn't know I had learned this and actually thought I was doing something wrong until it was explained to me. Essentially you are living in a modal world, where your context are the major and natural minor scale. The natural minor is perceived as that same scale but starting two notes below. It doesn't work that well for modern music.
About a decade ago I learned to recognize pitches as pitches. Previously G was the pitch that I sang as Do in G major, Mi in E major, Sol in C major etc. It's like recognizing the butcher, your neighbour, they guy in the front pew on Sundays, but never realizing it's always Jack who has his own personality. G is G. Wow.
A member in my family has "perfect pitch" though I'd call it "pitch recognition". It is like the rest of us know that an object is red or yellow. In the colour analogy, those colours also have shades. He will know that a pitch is B, and that in the context of A = 440 that B is a tad sharp. This world is the opposite of mine.
I have been told that singing pitch names in Solfege (fixed Do) would give me both worlds, but I don't know why that would be so.