I'm well aware concatenation wouldn't necessarily improve the strength. However, the idea is, even if SHA-1 was hopelessly broken. CONCAT(SHA1(x), SHA256(x)) would be at least as strong as SHA-256 (where "at least" means it may have the same strength).
If you know that it's a concatenation, couldn't you only look at the SHA1 part and completely bypass any other strong hash? On second thought probably not, because you might find any possible collision, that isn't a collition on all the other hash algorithms. If you bruteforce through a password list it would still apply though.
This doesn't work for collision resistance attacks. git commits aren't password hashes. Specifically, the attacker's goal in this case is to find different values a and b for which hash(a) = hash(b), rather than finding a value of m in h = hash(m) for known h.
Things like signed commits would still use the full hash, so that would make tampering with that impossible.
This solution would basically just make the UI backward-compatible while still requiring the complete modification of the internal to change the hash function.
You'd still risk a collision if you refer to commits using a shortened hash outside of git but something tells me that you don't even need a vulnerability to take advantage of that if you have an attack vector. For instance github seems to use 7 hex digit in short hashes, this could probably be bruteforced relatively easily (be it for SHA-1 or SHA-256). To give you an idea I looked at the current bitcoin difficulty (which AFAIK uses two rounds of SHA-256 internally and works by bruteforcing hashes with a certain number of leaning zeroes) and the hashes look like this: 000000000000000000028048b31e42bd53d3b36da90d1a840ae695ec1a5ee738
This would help if you _only_ shared the prefix, however git would still use the full hash.
The proposed method would have the advantage of keeping existing known abbreviations, which are _already_ less secure than SHA-1, while keeping the security of the second hash.
It also has the disadvantage that the full hash would become excessively large and unwieldy, so pros and cons.