You are adapting the NPC rules for character creation so you are essentially in homebrew territory here. The rules for creating PC and NPCs are completely different and adapting the NPC rules for use with PCs is going to involve at least a bit of modification on your part.
As part of this modification, it would be completely within your purview to determine what alignment the PC could/could not be and if there were any restrictions that went along with that.
This also fits with Rule 0 in D&D 5e which is that the DM has complete authority to run the world and to interpret the rules as they see fit.
As a note, no other official PC race to my knowledge has a strict alignment requirement. At most they provide guidelines for how members of the race usually are. So, I consider that approach when tackling this.
The alignment in the Monster Manual is not binding
The Monster Manual has this to say about the alignment of monsters:
The alignment specified in a monster’s stat block is the default. Feel free to depart from it and change a monster’s alignment to suit the needs of your campaign. If you want a good-aligned green dragon or an evil storm giant, there’s nothing stopping you.
So even if you decided to hold strictly true to the Monster Manual entry for describing how skeletons’ traits are, you still have wide latitude to have the PC choose what alignment to be.
I mentioned above how the rules and thought behind the building of NPCs and PCs were different and there is at least one instance of this that you should be very cautious about when trying to adapt this race.
If you want to take an NPC stat block and adapt it for a specific
monster race, apply the ability modifiers and add the features listed
in the NPC Features table.
Note that the process for creating the NPC based on this table assumes you are already starting with an NPC stat block and then basically dropping a monster template on top of it.
Thus, the features listed in the table are not intended to be used as simply standard racial traits for a member of this race. Instead it is how to modify another stat block to be part of the race.
For example, for the skeleton it lists the ability modifiers as:
+2 Dex, -4 lnt, -4 Cha
You should highly consider not basing the ability score modifiers you give your PC race on this. Those negatives are huge and would really really set a player back. Especially considering that your player wants to play a paladin, that Charisma negative could hamstring their entire character.
Note that very few PC races (and no standard PC races) give negative ability score modifiers to any ability. The reason was that the designers determined that players did not find it fun to have those. In this case, it would even go beyond that to making this character idea mechanically less rewarding for the player.
So, when you are making this race, you should find some other basis for deciding the ability score modifiers. Choose something that makes sense, but will still be fun for the character and puts them around the same power level as the other races already in the books.
I will also add that the NPC feaures table also says that NPC skeletons:
can’t speak but understands the languages it knew in life
I don’t think I need to tell you how inconvenient and pretty not-fun having a PC that can’t speak at all would be. Especially for a paladin.