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This was ''Living With Dinosaurs''
which was written by Anthony Minghella
of Truly Madly Deeply and The English Patient fame. It was a TV special from Jim Henson Productions which was made for Channel 4 and shown around Christmas 1990. The dinosaur was a plush toy that came to life in a Maybe Magic, Maybe Mundane fashion, a bit like Hobbes, though I have a feeling that unlike Hobbes ambiguous nature it was quite strongly hinted that this was due to the boy's imagination. The protoganist had a crippling phobia of fish, which is why the freak out when the dead fish fall on him is significant.
The Muppet Wiki article linked above says that there was an intention to repackage this special for showing in The Jim Henson Hour, but following that show's cancellation it was eventually shown in the US on Nickelodeon in 1993, which is where you may have seen it.
Oh, and the dinosaur (named "Dog") was adorable. Here's a clip:
https://www.youtube.com/watch?v=U3Bguq52cKg

I remember some TV show (or maybe a movie I forget) in The '90s where this kid had a puppet dinosaur as a companion. At one point the dinosaur knocks over this container full of dead fish on him and he freaks out. Also at one point the dinosaur keeps singing "I'm gonna bite you". It seemed really dark for a kid's show. That's all I remember.
Edited by Bootlebat