[RPG] How to validate the Spell Selections in an Eldritch Knight Spell List

This looks to be entirely up to DM discretion.

There are two equally valid positions on this and the rules do not refute either one.

The first is that this is a completely separate spell, and slot, and that the two have no overlap with your casting class. The spell you learned and the slot you obtain are kept separate from the class spells if you have levels in that class.

The second position is that it is a class spell, from your class list and you can retrain it as you would when you take a new level in that class.

The second position is mildly problematic in that you can continue to obtain higher level spells with the retraining, and it may lead to this MC being simply the obtaining of another 1st level spell slot, rather than feeling like a mini MC.

The first position is problematic in that it introducing additional bookkeeping and may be perceived to be overly harsh and pedantic.

In my games, I think I would allow the MC feat to simply be an additional first level spell slot and allow the spell to be retrained if it's from a class that allows that and you take levels in that class. I would probably not make them remember which spell goes with their MC beyond that the extra slot must be a spell from that class. This may be too lenient for some games though.

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.)