Classifying Polypeptides (and/or Proteins)
You can certainly refer to short peptides by their sequence. I don't know of any exact boundaries, but I've seen tripeptides referred to by either their three letter codes (Ala-Asp-Asn) or even the chemical name (alanylaspartylasparagine) although obviously that gets ridiculous pretty quickly.


  As the largest known protein, titin also has the longest IUPAC name of
  a protein. The full chemical name of the human canonical form of
  titin, which starts methionyl... and ends ...isoleucine, contains
  189,819 letters and is sometimes stated to be the longest word in the
  English language, or any language. However, lexicographers regard generic names of chemical compounds as verbal formulae rather than English words


For more than around 5 amino acids, just the sequence makes more sense. It would be interesting to analyse scientific paper abstracts to see how many have the sequence as the 'name', but anyway.

When we get to proteins, it would be unhelpful to refer to them by their sequence. They usually have names - enzymes are often named by what they act on (like 'alcohol') and how they act on them (like 'dehydrogenase'). In addition, there are structured names called E.C. numbers - sarcosine dehydrogenase is 1.5.8.2, for example.

As you point out, there are a very large number of sequences that do not correspond to proteins found in an organism. To name one of these is tricky, and if someone synthesises one then they would have to come up with some naming scheme to refer to it.