The hallmark of roguelikes is procedural generation, permanent death and no overarching progression, like the titular game Rogue. Roguelites (I know, one letter difference who cares) have overarching progression lines that make the game easier as you play it. Extreme difficulty as part of the roguelike formula is expected, since runs where you don't become very powerful very quickly happen, and the games are often balanced against some of the better possible runs.

Gungeon is punishing, but I regret to inform you you are FAR from too weak to finish floor 2 with the default kit of any character. Bashing your head against a challenge is fine as long as you're learning something every time you don't make it through, gungeon is simply a game where getting hit at all is really bad.

All that said, gungeon isn't the highest quality roguelike or roguelite. That's probably SLASH'EM (roguelike) or Elona (Roguelike(?)), and Hades (roguelite) or Windblown (roguelite, still in development). I won't even recommend roguelikes without graphics because the average age in this thread seems mighty low.