The book Fizban's Treasury of Dragons has been released. While it does contain some player-oriented material (2 new subclasses), something it doesn't do is update the Draconic Sorcerer to have additional Draconic Ancestry options for the new types of dragons introduced in the book.
If a player wanted to play a Draconic Sorcerer descended from one of these new dragons, would it be overpowered to let them draw their Draconic Ancestry damage type from that dragon's abilities? It should be noted that these types include types not otherwise available to the Draconic Sorcerer, such as Psychic, Thunder, Radiant, Force, and Necrotic.
Would being able to apply their Draconic Ancestry to these damage types break anything? I could potentially see issues if a multiclass Amethyst Draconic Sorcerer/Warlock could apply their Charisma bonus to the damage of their Eldritch Blasts twice, for instance.
Best Answer
Overall, this is probably fine as a "patch problems if they arise" implementation.
I've been through the sorcerer spell list for all of the spells of the relevant damage types (see this DDB filter), and besides the magic missile example cited below, nothing jumps out as a source of power imbalance.
The other obvious line of inquiry here is the interaction with these damage types and the damage resistance portion of the 6th level Elemental Affinity feature:
None of the new damage types introduced are any more abundantly common than the official ones. If anything, they are going to be less common than the ubiquitous fire and cold damage types.
The overall effectiveness of being able to gain resistance to these damage types is going to be situational and adventure dependent. If you find yourself in an adventure arc where one of them is abundantly common, then obviously being able to resist that damage type at will is going to be a huge boon, but this is a problem with the official damage types offered, so it doesn't really present a new concern, in my estimation.
Given these things, I think this will work just fine, patching any issues that arise (as I propose for magic missile). What I have not considered is multiclassing, and the question cites one such example where it could be a problem:
Sure, this makes the best cantrip even better, but this is not as egregious as magic missile since each beam of eldritch blast gets its own damage roll.
As a DM, just approach the implementation by having this conversation with the player:
And more importantly, we want to be sure that the player isn't going to be making character advancement choices based on something you might veto as the DM:
Magic Missile becomes wildly powerful for an Amethyst Draconic Sorcerer.
Magic missile deals force damage, and as explained in this answer, magic missile only uses one damage roll. Now, the Draconic Bloodline sorcerer's 6th level feature Elemental Affinity states:
Magic missile has one damage roll. So a 20 Charisma sorcerer casting magic missile at 1st level would deal 3*(1d4+6) (average 25.5) force damage automatically, no attack roll or saving throw required. Upcasting, this free damage scales all the way up to 11*(1d4+6) (average 93.5) when cast at 9th level. Averaging 25.5 damage from a 1st level spell, that hits automatically, of the least resisted damage type (force) is completely broken.
Combine this the sorcerer's metamagic, and things just get more unbalanced. The sorcerer can use Empowered Spell to reroll 1s and 2s on the damage die for magic missile, or with Distant Spell, they can extend the range of their free damage to 240 feet.
Obviously, this can be patched by just ruling that the darts of magic missile require separate damage rolls.