It isn't an assumption; the null hypothesis here is that that
doesn't happen. Genes coding for cryoprotective proteins have indeed, as you suggest, evolved independently among various species. The resulting genes, despite all producing proteins similar enough to do the necessary job, are "radically different" and "highly diverse." [1]
What's different in this case is that, in three otherwise very distantly related species of fish, we find their antifreeze proteins are coded for by the same genes:
> But, the isolated occurrence of three very similar type II AFPs in three distantly related species (herring, smelt and sea raven) cannot be explained by this mechanism. These globular, lectin-like AFPs have a unique disulfide-bonding pattern, and share up to 85% identity in their amino acid sequences, with regions of even higher identity in their genes. A thorough search of current databases failed to find a homolog in any other species with greater than 40% amino acid sequence identity. [1]
In light of the fact that all other genes known to code for these proteins are very distinct both from this one and from one another, that three species should have a near-identical sequence coding for a near-identical protein suggests rather strongly that this version of the gene arose in one species and was then acquired by the other two, i.e., that horizontal gene transfer has occurred among these vertebrates.
[1] https://pubmed.ncbi.nlm.nih.gov/18612417/
What's different in this case is that, in three otherwise very distantly related species of fish, we find their antifreeze proteins are coded for by the same genes:
> But, the isolated occurrence of three very similar type II AFPs in three distantly related species (herring, smelt and sea raven) cannot be explained by this mechanism. These globular, lectin-like AFPs have a unique disulfide-bonding pattern, and share up to 85% identity in their amino acid sequences, with regions of even higher identity in their genes. A thorough search of current databases failed to find a homolog in any other species with greater than 40% amino acid sequence identity. [1]
In light of the fact that all other genes known to code for these proteins are very distinct both from this one and from one another, that three species should have a near-identical sequence coding for a near-identical protein suggests rather strongly that this version of the gene arose in one species and was then acquired by the other two, i.e., that horizontal gene transfer has occurred among these vertebrates.
[1] https://pubmed.ncbi.nlm.nih.gov/18612417/