1st Off: I hope your fish gets better; I really do. Whether it's merely the result of anthropomorphism/projection, the unconditional love I derive from animals is nevertheless very real to me. And sometimes, "real to me" is real enough.
As to whether or not it's normal, I cannot comment--as I've little business assessing what is/isn't normal--, but I certainly get very attached to animals. My last experience with a pet's death included me falling to my hands and knees while at the animal hospital, uncontrollable sobbing... pretty much a break-down, basically.
So, while you may not be 'normal,' you certainly are not alone. I feel as though I get along easier with animals than I do with many people, often. And yes, I regularly distance myself emotionally from what others are going through. 'Cause if I start to seriously reflect on the sadness/illness/what-have-you of another, I cannot stop. It is as if I feel it too much. And there's so little I can do, I just feel helpless...
Yeah...
I'm an emotional guy. If I were to show my heart on my sleeve to the extent that I might like to, I fear that I would be perceived as abnormal. Guys aren't supposed to be emotional, according to some. So I keep to myself, mostly. But when I do get emotional, I tend to "go all out." I could say that "it sucks." A better, more constructive and realistic thing to say would be that "it is."
Once more, I sincerely hope that your fish's health improves and you can return to enjoying one-another's company.
Kind Regards to All.