First thing I’d like to say is that one must not try to search too deep into song lyrics.
That being said tou are absolutely right in saying that un autre que moi et un autre moi mean different things. And the song does say un autre que moi.

• Un autre que moi refers to another physical person.
• Un autre moi refers to another abstract self, same body but different personality.

You are not linking to any translation but I’ve found this one:

And they confess
I’ve never said that
I find someone other than me
We find ourselves in an terror attack

that does not look too bad. As you see the translation is different from the one you are mentioning.

The first translation I’d hit upon went like this:

Let them approve
I never said that
I’ve discovered another than me
We find ourselves in a fight

both translations agree about un autre que moi but they are on the whole different and I think the first one I’m pointing to is much better.

Don’t forget the translations on these sites are usually made by non specialists, there are often errors, and with song lyrics it’s the same as with poetry, you need to do a lot of interpretation.

Concerning the particular words you are pointing to I would definitely go for “another than me” or “someone other than me” because “another me” would really change the meaning too much.

Plainly, it means “someone else than me” .

The lyrics are duplicated below for easy reference.

Élancée comme une panthère dans la fumée
Le regard noir, le visage fier, le pas feutré
Ce qu’on a découvert sur terre m’a fait pleurer
Oui j’avais l’air et la manière d’aimer

Élancée comme une guerrière dans la mêlée
Le regard fixe, prête pour la rixe, j’ai riposté
Le goût du danger sur mes lèvres l’a fait pleurer
Il avait l’air et la manière d’aimer

Qu’ils nous retrouvent
On n’se rendra pas
Et qu’ils approuvent
J’ai jamais dit ça
Je découvre un autre que moi
On se retrouve dans un attentat

Tout allait de travers
J’sentais monter dans l’atmosphère
Un coup d’éclat, d’éclair, un coup d’tonnerre
Droite, gauche, bras d’fer, on se défend
Mais t’as vu la gueule du sentiment ?

Qu’ils nous retrouvent
On n’se rendra pas
Et qu’ils approuvent
J’ai jamais dit ça
Je découvre un autre que moi
On se retrouve un autre attentat

There is no evidence to infer wheter “un autre que moi” means “someone other than me” or “another me”, “another myself”. There are two main protagonists in the situation referred to, a woman, who is the narrator (first person point of view), this being elicited from the gender agreement (élancée) and a man, this last fact being elicited from the masculine pronoun “il”. In such a context of vague references I’m inclined to opt for the most straightforward interpretation: there is no question of another self but only of someone else (il).
There is an undubitable parallel: she cries, shows love in one way, he cries too and shows love in the same way, but from that to infer a “double value” for “un autre que moi” is doubly contrived, too complicated, so much so as it is just as legitimate if not more to expect “un autre moi” (another one such as me”), therefore someone different yet not unlike myself. We are not even sure that the masculine protagonist is on her side; the speculation of finding him on the other side is open. I do not personally understand the aim in making in this and countless other lyrics, the meaning so unaccessible to the listener, so uncertain; the impact is obviously enormously diminished by this intended vagueness as the audience is bound to retain nothing specific from such a piece of literature, even after much pondering over of the few possibilities.