If another context is sufficiently different, it would be fine to use a controlled fixed-length binary-vocabulary there. (But, the ASCII bytes for "sha1:" would still be a pretty robust and largely self-documenting choice.)
the important thing here is standardizing the encoding, i.e. there should be a valid repr in {binary, hex, b32, b58, b64, ... bemoji ...}
so that, say, if your input field _only takes hex_ you have a way to enter the hash. this problem surfaces when tools expect reprs to be in a particular encoding (all too common), and you dont have power or access to make the tool better (sadly all too common too).
I'm not sure I see the point.
I deal with outputs from crypto functions on a daily basis. Never once have I intentionally rendered it as 8-byte ASCII. It's most often in hex. Mixing and matching encodings (e.g., prefixing a hex string with an ASCII string) for a single blob of data is silly and just causes headaches whenever you need to change encodings, either by having some data double-encoded or by having to specially handle the prefix separately.
Compared to that, adaptation to other constrained systems is a case-by-case issue. And if those systems are already capable of squeezing in these non-native slightly-longer hash-like-strings, then a few more bytes usually won't hurt, or if they do then whatever deep-in-the-bits coder (like yourself) who's shoehorning things in can handle compactification. The display/exchange format should be as casually readable as possible.
Such an ASCII-name-inspired prefix is readable in all encodings. In some, it's just a magic number (but one that any coder can make educated guesses about); but in the one encoding that's likely most important for mutual comprehension between humans (user-exchanged strings and URLs), it's super-duper-readable.
And yes, the prefix should in general have special handling: as the multihash project README notes in an important "warning", the prefix lacks the same distribution as the other bytes. Treating the whole thing as an opaque-but-still-reliable identifier invites indexing misoptimizations right off the bat, and then other later bugs, if any of the hash functions become deprecated, or a new hash is added with variant semantics.