Arrow of Slaying is usable as a non-attuned item and has a DC of 17.

Rod of Lordly Might must be attuned and has a DC of 17 for its abilities.

Would it be highly unbalancing to have an attuned item's DC attach to the character's spellcasting stat? For non-casters it could attach to CON or their highest mental stat? Intelligent weapons with ability DCs would use either the weapon's stats or the attuned character's. Non-attuned items are created with the idea that anyone could use it at the creator's power level, but attuning an item to a character would mean that the magic is entwining itself with the bearers 'spirit'. This would also affect to hit modifiers as the item is an extension of the character's powers.

I am sure this idea has been around the block a multitude of times, but I have not seen any threads that ask if it is unbalancing.

Unbalanced? That would depend on the item in question. If a Wand of Fireballs keyed off of caster ability then yes, that would be unbalanced. A DC of 15put it right smack dab in the middle between the lowest (DC 10) and highest (DC19) that PCs can reasonably expect. For a lower level party a DC 15 makes something like a Wand of Fireballs a sweet boost that might allow them to overcome an otherwise unbeatable enemy. For a higher level party it means that it’s still useful, but not the equivalent of having 7 free additional 3rd-level spell slots. That’s because if they were to just cast it using their own spe slot it would be more effective. It allows the DM to give them a significant shot in the arm when the story calls for it without having to worry about such a negaweapon staying that powerful forever.
     (When that PC could cast Imprisonment, [Tooltip Not Found], or Psychic Scream or possibly even Fireball at DC 19 for themselves, a Wand of Fireballs is a backup piece for that PC when they’re out of better options. Or if they just need to take are of some light work that Wand is about right.)

A Rod of Lordly Might is already really super powerful. If that could go to DC 19 that would be Artifact level. It doesn’t really have anything to do with Attunement either. An Arrow of Slaying doesn’t require Attunement because it’s basically a consumable like a potion or scroll. Once it works it works. If you miss with it you can recover it, but once it blows an extra 6-60 damage into something it’s no longer magical.

The whole thing with Attunement isn’t so much that a user becomes “spiritually entwined” with the item, more just gotten to know each other well enough to work together. So when a character uses a Wand of Fireballs or the RoLM, it isn’t that “they are now ‘one’” or anything. If that character uses that Wand, the wand still puts all of the gusto into it for the PC which is why magic items only require concentration for spells they cast and not spell slots or components.

The logic behind them is that things like Spell Attack and Save DC are imparted into the item at creation and reflect a measure of the creators ability at that time. Back in the day it was possible to get multiple Wands of the same type that had different numbers of charges and different DCs. But imagine the bookkeeping pain in the neck having three Wands of Fireballs, one with DC 14 & 5 Charges; another DC 16 & 7 Charges, and third with DC 18 & 9 Charges! So they scrapped all of that and made it flat this edition. (And it’s muuucch better this way, trust me.) However, if I had a player with a PC who wanted to make their own Wand of Fireballs, I would allow it to use their Save DC at construction if it is higher than 15, but theirs would be a 1 of a kind and Attuneable by no one but that PC as long as she were alive. That way it would be possible to lose it somehow but neither sell it nor give it away except under the most special of circumstances or something. I gift to a Dragonlord? Sure, one for every party member? I say nay nay.

There are already official magic items like this; the Staff of Charming for one. Of course, it specifies that it must be attuned by certain types of spellcasters, while the Rod of Lordly Might does not.

(Also note that the Arrow of Slaying is a consumable item, as well as geared towards a particular target type not the user, so it'd be mean to make it attunement.)

It seems to me that the important things are 1) how general-use the item is, and 2) the item's rarity.

If an item could be used by a non-spellcaster, then it wouldn't make sense to use their "spellcasting" DC. Even items that must be attuned by spellcasters in general can be tricky too, since some races come with spells and a spellcasting ability, which a player may not be leaning into; it wouldn't be very fair to make a perfectly decent magic item "suck" for them.

As for rarity, a common, uncommon or rare item has a DC set to a value appropriate to the rarity; otherwise the item would be far too good for anyone with a decent spellcasting ability. When you get into very rare, legendary and artifact items, then it becomes more appropriate - these are geared to particular spellcasting classes, or unique. Especially for special items like the Vestiges in Critical Role, it makes sense for them to "scale" with the player's ability. But even with a unique legendary item, you might want to set a particular DC - it'll be pretty high anyway, and possibly higher than the character's DC!