Yes, both effects can apply to the same spell.

There's nothing in the wording of either feature to indicate that it excludes other features adding damage as well. Many other combinations of features can work this way -- a barbarian's damage bonus from Rage is cumulative with their Strength damage bonus and other bonuses such as the Dueling fighting style. The word all of these have in common is add.

Contrast this with Thirsting Blade, which says:

You can attack with your pact weapon twice, instead of once, whenever you take the Attack action on your turn.

It doesn't say "you can attack one additional time", so the reason it doesn't stack with Extra Attacks is that neither one adds attacks, they both just change the number of attacks from one to two.

This is also similar to how the barbarian and monk Unarmored Defense features don't "stack" with armor. Armor provides one way of calculating AC, Unarmored Defense provides another. A given character can only use one.

So, in general, effects or features that let you add something to a roll are cumulative, but effects or features that change a value are not.

It will make them stronger, but not much

This answer is about Wizards, but I agree that more spells from the same list are not much improvement. Usual Sorcerers take the 15 spells¹ they find best or strongest, so by definition the next 5 will be weaker.
They have the same number of slots, and more importantly, the same number of actions per encounter.

Example: Lightning Bolt

In most situations Fireball is better than Lightning Bolt, as it usually covers more area. It effects more enemies, so most Sorcerers only know the former as they can't afford both.
After this change they might be able to take both, but in the vast majority of the cases, they will still chose Fireball to cast. Only in the few times when the enemy is lined up nicely, or is resistant/immune to fire but not to lightning, will they cast Lightning Bolt. So in about 10% of encounters, they will not lose 50% of their damage (they will lose some damage, as most likely less enemies fit into a line than a sphere).

This comes up to 5%, hardly unbalancing².

Spells from other lists

This is where the idea can go wrong, so you have to examine it on a case-by-case basis:
A twinned Tasha’s Hideous Laughter or heightened Evard's Black Tentacles³ is very powerful, but no metamagic will make Flame Blade great.

This just can't be answered unless we see the whole list for all archetypes, but than it might be too broad.

Try to fit all possible metamagics on the spells, and if it still does not look overpowered, you are fine.

^{1. Sorcerers know 15 spells on level 20.}

^{2. The difference is even less, you could just cast something completely different like Toll the Dead or Haste.}

^{3. I think named spells (Evard's, Tasha's) are usually created stronger, because they are unavailable for Sorcerer's. Consider them a Wizard class feature compensating for no metamagic}