Examples of an undoubtedly correct sentence would be:
"He has been studying English ever since he started at secondary school five years ago."
"He has studied English on and off since he started at secondary school five years ago."
Personally I think the above sentences are OK without "he started at secondary school" (and then also without "ever"), so "he has been studying English since five years ago" or "he has studied English on and off since five years ago" would both be fine. Others may disagree??
However "he has been studying English for five years" would be more common, and possibly more natural (??), than "he has been studying English since five years ago".
Likewise "he has studied English on and off for five years" is possible, though actually I prefer "he has studied English on and off since five years ago", because "for" may imply continuity, which is in tension with "on and off".
If my examples are correct, then the biggest problem with your example is that you wrote "has studied" instead of "has been studying". Although "he has studied English for five years" is more natural than "he has studied English since five years ago", I would also prefer "he has been studying English for five years" rather than "he has studied...".
I think the reason why "he has studied English since five years ago" is particularly unnatural is that "he has studied English since five years ago" makes it sound as if studying English was either a one-off event (or a possibly short-term occurrence which lasted less than 5 years) or perhaps a perfectly continuous event of 5 years with no breaks.
Consider: "We know that until five years ago, he had never killed anyone. But we also know that at some point since five years ago he has killed someone."
Likewise: "We know that five years ago he was unable to read or write. But since five years ago he has learnt to read."
Likewise: "He has stood in front of my door since two hours ago, glaring menacingly at all the passers-by."