Pushing my DM limits...

I want to start by saying this is super cool, and you are awesome for wanting to figure this out for your player. However, this will be a tough road to balance out, haha. Ideas time!

From a mechanical perspective the biggest issue is the action economy that the player would be allowed. In most scenarios presented in this thread it is a Rogue/(Fighter or Barbarian) as examples. Even at level one as long as the Fight/Barb is within 5 ft of the enemy the Rogue is likely looking at 2d6+3 (sneak attack from nearby ally) for damage in a round, and if the Fight/Barb can attack as well then it is adding on 1d8 to 2d6 + STR as well. Even under the multi class method the ability to divide and conquer is a lot.

I do think that the multiclassing method is the closest to getting things to be balanced, but the actions available per round need to be addressed. Before that thoguh, I think that the HP of the twins works out in multiclassing as the base numbers should be looked at without CON mods involved as that will be dependent on player preference and stat scheme. Fighter with no CON is 28 and Barb with no CON is 33 at the standard stat progression for 4th level. The twins would be 29-32 depending on Fight/Barb choice, but that is also assuming that set HP at each level is used, but that should be the rough average, blah blah MATH!
The other idea I thoguht of was to use the Druid animal companion as an example, and that is eluded to in DM Kiado's thoughts as well. However, that isn't really a thing in 5e, haha. I assume for the same action economy reasons that I brought up. The Ranger doesn't even have the option until level 3 where you get a CR 1/4 animal companion ... so yeah, not that helpful of an example.

I think that a certain set of actions available is the way to go, but at the same time that feels very limiting in certain scenarios or not limiting enough. If they are force into having only the standard set of actions then one is not attacking, or one isn't moving, or one can't react if an enemy were to run in between them, etc. What could help in combat might be to allow the Help action to be given in conjunction with the twins. Instead of each getting an attack on the enemy the person attacking gets the Help action (PHB pg 192) for free, and also looking at the Working Together (PHB pg 172) could help for ideas for out of combat synergy between the characters. I think a range restriction on how far they are apart they are could be interesting as well.

Also, after playing with your proposed stat scheme you will probably have to bump up the "neither character can have more than 3 scores over 8" given that the average roll of 2d6 is 7, and after doing three full sets of 4d6's and 2d6's I might have one that even has enough stats that low. That same set would also only have a high of 14 out of all the rolls. I honestly can't think of a good way to disperse the stats without it being overpowered in some way. Playing with the standard stat array falls apart when you have one twin good at a and b while the other can do a and c. A potential work around could be having the low stats rolled as 1d6+3 to allow for marginally lower stats, but this still makes that half of the twin set really poor at certain things obviously with a guaranteed -1 to -3 in half their stats. Then again if the pair is truly doing what one can and the other can't then I think it could potentially work out still.

So, all of that to say, it is definitely a hard road ahead in terms of balance, but I think it could be done. I don't know that it can be done from a purely academic/theoretical stance, but will likely need some play at the table before it could be ironed out well. So hopefully some of what I have typed is helpful!

P.S. One last random thought: you could potentially do a homebrew class as well to help the balance issue.