If they fail the check to find the trap, they don't even know anything's there. Hard to identify and classify something you haven't seen and don't know exists.
Do you mean fail to disarm the trap?
No worries, mate. :) I don't think there's any mechanical advantage to knowing what kind of trap something is, so I don't think it makes a world of difference in terms of actual gameplay. Whether they can put a name to it or not, they still can't open it on that try.
But remember, for what it's worth, what's happened in the campaign isn't that character's entire experience. They could have tinkered with these when they were first learning about traps. They could have read about them in a book. Heck, they could have heard a bard telling a story that involved one once when they were a kid, and they connect the dots when they see this one in front of them and put a name to it.
So, it's possible that they'd be able to place a name to it - but it's a moot point. Since they failed the Disable Device check, they can't open it on that go - they'll have to try again, and there's no bonus for being able to put a name to it.
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The mechanical advantage to identifying the type of trap would come into play if other party members have immunities.
Say the Rogue finds a trap on a chest and identifies it as a poison needle trap. That would be useful to know because another character might have poison immunity, or a spell could be cast that confers an advantage (like a resistance or bonus to save, etc.).
If it were me, I would allow the Rogue to identify the type of trap if the Rogue beat the Perception check to find the trap by 5 or more and had encountered that specific type of trap before. Even then, I would only allow the Rogue to identify trap in general terms; I wouldn't, say, allow them to identify the type of poison used to construct the trap, or the caster level of a magical trap, etc.
-Skeld