I think they are pets but not in the same sense as a cat or dog. I have to be honest, I would never be attached to my fish the same as for example my cat. I was sad when i lost my 11" oscar and the previous 8" dempsy because I had cared for them for so long.
I think for me its more a hobby in the sense that I am trying to do my best to give some confined innocent creatures the best life I can. People say fish dont have feelings or feel pain...etc. I can genuinely see when they are happy. Almost like having a puppy or kitten that is happy and playing.
My goal is to eventually have a very big tank instead of the 3 smaller ones and give them all more swimming space and more freedom. People hate my tanks because they say they cant see most of my fish and that its not very bright but I know my fish like the security of all the hiding spots and the fact that there are no flood lamps above my tank