What is the relation between renormalization in physics and divergent series in mathematics?

You are conflating three conceptually different categories of "regularizations" of seemingly divergent series (and integrals).

The type of resummations that Hardy would talk about are similar to the zeta-function regularization - the example that is most familiar to the physicists. For example, $$S=\sum_{n=1}^\infty n= -\frac{1}{12}$$ is the most famous sum. Note that this result is unique; it is a well-defined number. In particular, that allows one to calculate the critical dimension of bosonic string theory from $(D-2)S/2+1=0$ and the result is $D=26$. Fundamentally speaking, there is no real divergence in the sum. The "divergent pieces" may be subtracted "completely".

However, in the usual cases of renormalization - of a loop diagram - in quantum field theory, there are divergences. Renormalization removes the "infinite part" of these terms. A finite term is left but the magnitude of the term is not uniquely determined, like it was in the case of the sum of positive integers. Instead, every type of a divergence in a loop diagram produces one parameter - analogous to the coupling constant - that has to be adjusted. Because the finite results can be "anything", this is clearly something else than the zeta-regularization and, more generally, Hardy's procedures whose very goal was to produce unique, well-defined results for seemingly divergent expressions. Infinitesimally speaking, the Renormalization Group only mixes the lower-order contributions (by the number of loops) into a higher-order contribution.

So these are two different things that one should distinguish.

There is another category of problems that is different from both categories above: the summation of the perturbative expansions to all orders. It can be demonstrated that in almost all fields theories - and perturbative string theories as well - the perturbative expansions diverge. For a small coupling, one can sum them up to the smallest term, before the factorial-like coefficient begins to increase the terms again, despite the $g^{2L}$ suppression. The smallest term is of the same order as the leading non-perturbative contributions.

At the very end, if the theory can be non-perturbatively well-defined - and both QCD-like theories and string theory can, at least in principle - the full function as a function of the coupling constant $g$ exists. But it just can't be fully obtained from the perturbative expansion. The Renormalization Group won't really help you because it only mixes the perturbative terms of another order to a perturbative diagram you want to calculate. If you don't know the non-perturbative physics, the equations of the Renormalization Group won't fill the gap because they will keep you in the perturbative realm.

So I have sketched three different things: in the Hardy/zeta problems, the answer to the divergent series was unique; in the particular $L$-loop diagrams in QFT, it wasn't unique but the infinite part was subtracted and the finite part was obtained by a comparison with the experiments; and in the perturbative expansion resummed to all orders, the sum actually didn't converge and indeed, it didn't know about all the information about the full result for a finite $g$.

The last statement may have some subtleties; at least for some theories, the non-perturbative physics is fully determined by the perturbative physics. But I think it is not quite general and we have counterexamples - e.g. for AdS/CFT with orthogonal groups and different discrete values of $B$ etc. So it means that the perturbative expansion doesn't uniquely determine the theory non-perturbatively.

Because the three examples differ at the level of "what can be calculated" and "what cannot", they are different.

There are overlaps between QFT renormalization problems and mathematical approaches to divergent series summations (see for instance the section 'physics' in the Wikipedia article $1 + 2 + 3 + 4 + \dots$ ) However, most of the infinities encountered in modern physics are 'harder' than those that can be attacked by existing divergent series summation methods.