Your main point below is not entirely correct. No Exit begins with the characters describing themselves to each other - thus establishing their identities in their own words. This first identity is, however, nothing but a lie.
Garcin is the first to reveal his "true" identity, but again this identity is established through his version of himself, not the others. The other characters then reveal their true identities.
The concept that identity is defined by others only emerges near the end when Inez tortures Garcin by calling him a coward. Here we see identity being determined from an outsider, conflicting with the "self identity" that Garcin has of himself. Interestingly, Garcin refuses to leave hell in his efforts to alter Inez' perceptions of him. However, we also cannot forget that these perceptions are based entirely on Garcin's earlier descriptions of himself. So a key point is that he is stuck with two people who see him as he earlier defined himself to them. He is essentially stuck with his own view of his identity, and hell is therefore his inability to escape/alter his own identity.