[RPG] Can a wizard’s level-up spells choices be from outside their school

Extra Spell is specifically addressed in the FAQ (p. 40):

Can you take spells from spell lists other than your own with the Extra Spell feat (CAr 79)?

The Extra Spell feat allows you to choose a new spell, but it does not remove the restrictions of how you would normally pick your spells—so they must be picked from your own spell list.

This is the general rule -- you have a class ability that lets you cast spells from a particular class list only. For instance, the wizard has the following ability:

Spells: A wizard casts arcane spells which are drawn from the sorcerer/ wizard spell list.

For something to give you spells not on your class list, it has to explicitly call this out as an exception to the more general rule. The Master Specialist ability is worded almost identically to the wizard's regular spell book entry, so it clearly does not allow this:

When you reach 2nd level, you can add one spell of your chosen school to your spellbook. The spell can be of any level that you can cast, and it is in addition to the normal spells gained when increasing your level.

If it isn't on your class list, how would you even determine if it was a level that you can cast?

You can switch out any of the "any-school" spells

The PHB Errata, has now fully solved this issue.

From the "Rogue" section of the errata:

[New] Spells Known of 1st-Level and Higher (p. 98). In the final sentence of this subsection, the closing phrase has changed to read as follows: “… unless you’re replacing the spell you gained at 3rd, 8th, 14th, or 20th level from any school of magic.”

Similarly, the "Fighter" section of the errata states:

Spells Known of 1st-Level and Higher (p. 75). The final sentence ends as follows: “… unless you’re replacing the spell you gained at 3rd, 8th, 14th, or 20th level from any school of magic”.

This means you can indeed retrain the "any-school" spell and replace it with another "any-school" spell as appropriate, both as an Eldritch Knight and an Arcane Trickster.

As a historical note, the fighter section was added to errata at least as early as 2016 while the rogue section was not fixed until 2020 though this tweet from Jeremy Crawford already stated that the same reasoning should apply:

Like an Eldritch Knight, an Arcane Trickster can retrain their 1st-level spell from any school. (An unintended omission.)