I would need to know your background to be able to give a really informative answer, but I guess you could worse than Autour des conjectures de Bloch-Kato (Fontaine/Perrin-Riou) in the volumes Motives.

I would bet the question you want to ask is for $G$ the Galois group of the maximal extension unramified outside a finite set and that you want to consider continuous cohomology (then the answer is no, you cannot do that - the reason being that the dimension of a subspace of $H^1$ is bounded below by the order of vanishing of some $L$-functions, and that could be huge).

Interpreted literally, this cannot be true: congruences between modular forms of different levels (here an Eisenstein series of level $1$ with an eigencuspform of level $N>1$) will not be congruences for all $n≥1$ (the congruence will not hold for primes $\ell$ dividing $N$). With that restriction in mind, the answer is yes.

It seems to me to be the hardest case.

"Therefore, I am trying to understand what he means by "why" [...] why can't the existence of the group be a coincidence ?" You know what? I think your question is exactly the answer to your question.

Then again, if I were told in Japanese that my proof is "too difficult to understand" or that "lemma X seems difficult", I would immediately interpret this as my interlocutor trying to convey that my proof is faulty without upsetting social norms.