In the video you quoted, Mikhael Gromov said nothing about Algebras containing relations that are not computations.

The statement:

All traditional descriptions of mathematics, in my view, are greatly faulty

includes the description of Algebras as a particular case.

In your undergraduate mathematics studies, did you take the course series in group theory, rings and fields, and then algebras? It is from that viewpoint that I state there are algebras [...]

I totally agree, you derived the right consequence of that viewpoint. Nevertheless, according to M. Gromov, that viewpoint is wrong (I do not claim to have the right viewpoint neither):

All traditional descriptions of mathematics, in my view, are greatly faulty, and this is the reason why there is no model of mathematics realized in computers, why there is no model of understanding languages, because we have an absolute wrong perception traditionally build in development of numerical mathematics in logic distorted our perception of ourselves, of mathematics and of languages. It is as distorted as the Sun orbiting the Earth, [...] it is the wrong language, the wrong description.

M. Gromov, reference here.

Your conclusion

Mathematics contains more than just computation

which is justified from the perspective that you are talking about, seems to be equivalent to my conclusion that the whole point of mathematics is that it contains less than computation. In other words, the non-computational mathematics in your approach is the analog of computations from which the property of computability was ignored during a process of abstraction. Considering numbers, if you say that some real numbers are non-computable, I may equivalently say that these non-computable real numbers are obtained from the computable real numbers by ignoring the property of computability during a process of abstraction. Hence, non-computable real numbers are less than computational real numbers from a conceptual point of view: they are real numbers minus the property of computability. This conceptual subtraction is what makes mathematics so powerful: it allows shortcuts in irreducible computation.

Sure, the subgroups of A that are not related to computation are an abstraction of the computational notion of subgroup. My claim "mathematics is an abstraction of computation" does not mean that "mathematics is just computation". Mathematics is obtained from computation by ignoring some features. In this case, you are ignoring a feature known as computability.

The definition

"Mathematics is the language of measurement" -- Steven Krashen

involves a computation, since the act of measuring is a (concurrent) computation: the transmission of information from one system to another. To overcome the computational foundations of mathematics, it is important to develop a foundation of mathematics that does not involve computation of any kind. Then computation should be derived from such a hypothetical system without using computation in this derivation process. I do not know such an approach; maybe another person knows.