Identify does NOT reveal a curse, as per the rules. That's one thing it specifically does not do.
Most cursed items also double up on this reassurance by mentioning in their own text that Identify does not reveal the curse.
If your DM is revealing that an item is cursed to you when you cast identify on it, then they're doing it wrong.
In most cases in 5e, Cursed items must be attuned, and when they are attuned you can't voluntarily break your attunement to it.
