Thanks for the response. I am not 100% sure what you mean with "plain Git level". It is possible to check, if a local tag exists at the specified remote and if the tag is set on the same commit.

E.g. 

tag 0.0.0 is on the same commit, the others on different

the local repository has the tags of origin2

origin1 tags: 0.0.0 0.0.1 0.0.2

origin2 tags: 0.0.0 0.0.1 0.0.2 0.0.3

git fetch origin1 tag 0.0.0 -> success as the commit is the same

git fetch origin2 tag 0.0.0 -> success as the commit is the same

git fetch origin1 tag 0.0.1 -> rejected as the commit is different

git fetch origin2 tag 0.0.1 -> success as the commit is the same

git fetch origin1 tag 0.0.2 -> rejected as the commit is different

git fetch origin2 tag 0.0.2 -> success as the commit is the same

git fetch origin1 tag 0.0.3 -> couldn't find remote ref (tag is not present)

git fetch origin2 tag 0.0.3 -> success as the commit is the same

Another possibility would be:

git ls-remote --tags origin1

-> shows commits corresponding to tags of origin1

and 

git ls-remote --tags origin2

-> shows commits corresponding to tags of origin2

So the information is not stored locally by git, but could be stored and is available through git commands.