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:
At the same time, you can spend 1 sorcery point to gain resistance to that damage type for 1 hour.
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:
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.
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:
This will probably be just fine, but I reserve the right to fix broken things if they come up.
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:
When it comes time to level up and make character decisions, if you find a powerful interaction here, run it by me first so we can talk about it. If I'm going to veto something, I would rather do it after having a conversation with you about it before you have made any character decisions, rather than after you have made those decisions.
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:
Starting at 6th level, when you cast a spell that deals damage of the type associated with your draconic ancestry, you can add your Charisma modifier to one damage roll of that spell.
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.
