No. Git should never do that, it would make git worse.
There are a lot of other different metadata that you could imagine to store per commit, but git already supports storing arbitrary data in every commit, you don’t need special casing for some type of metadata, just store it in the commmit as everyone already does, and perhaps build your own tools on top of that if you have special needs.
As explained, the storage backend in git is pluggable but still not flexible enough.
There has been efforts to store git repos in torrents, and to store got repos on crypto blockchains, but all are big architectural challenges, for example people want everything to be backwards compatible for starters, and some want to be able to partially store some content somewhere else, while still keeping all existing use cases efficient.
There are a lot of other different metadata that you could imagine to store per commit, but git already supports storing arbitrary data in every commit, you don’t need special casing for some type of metadata, just store it in the commmit as everyone already does, and perhaps build your own tools on top of that if you have special needs.