As the question says, are there any such items? Specifically permanent ones, not temporary consumables, although items with charges that can run out are acceptable. I'm designing a magic item for a friend, who plays a tortle, and I wanted to give him a steam breath ability like the dragon turtle. I would also probably open it up to be used by anyone with a Breath Weapon ability. But, I wanted the item to be non-attunement, since I was going to make it kind of weak. However, if there is no precedent for this kind of item, then I would be very hesitant about making it. Are there items like this? And if not, would making such an item be a big power issue?
There are three magic items in the DMG (pg) with specific requirements
Apparatus of Kwalish (151) requires that you are Medium or smaller.
Horseshoes of a Zephyr (173) require usage by a horse or similar creature.
Horseshoes of Speed (173) require usage by a horse or similar creature.
Many magic items have implied restrictions
The ability to speak command words is one such restriction. There would be no significant balance issues requiring certain abilities like a breath weapon other than restricting who can use the magic item. I would recommend a small bonus to all who use it with the primary benefit being used by the characters in question; this way some discovery can be made for the player who decides that the weapon is most befitting a breath weapon user.
Spell scrolls are such items.
If the spell is on your class's spell list you can use an action to read the scroll and cast its spell [..] Otherwise, the scroll is unintelligible. (DMG, p. 200)