The current use of tag named "norm" is actually meant normed vector spaces. But it is not different from the tag "normed-spaces" Unfortunately, I cannot propose this as a tag-synonym due lack of reputation on this particular tag :).
On the other hand, this use can be misleading as we can equip many algebraic structures with a compatible metric to define their normed counterparts. In fact, normed groups, normed rings, normed fields, normed algebras (and many more) are useful and well established concepts.
So, here I have two proposals:
Merge them to make a tag "normed-vector-spaces".
I am well aware that people use normed space as a shorthand for normed vector spaces, but I don't think writing the full name would do any harm :)
Recycle the tag name "norm" in the inclusive sense to capture all the above mentioned notions.
Some of you may find this is unnecessary as we currently have no questions in this sense. At first, I was actually formulating a question about a topological semigroup whose topology is metrizable before I found this possible loophole. It will only be a matter of time until we receive questions about general norms.