@DxellaFrom a code standpoint, the MCV inherits its Faction from you as the player.
If you for example are playing as NOD, then capture a GDI Construction Yard, even if you lose your original MCV, should you move your GDI MCV (with you as NOD) and redeploy it, that MCV will become a NOD MCV.
The same applies to a GDI Player capturing a NOD Construction Yard. Soviet Player capturing an Allied Construction Yard. Allied Player capturing a Soviet Construction Yard. Once you move it, and redeploy it, it will inherit from whatever faction you are, not whatever faction it originally was.
Observation I noticed:
A Construction Yard that is on a Custom Map, that otherwise isn't faction aligned will not provide any Construction Options. If you lose your original Construction Yard, that captured Construction Yard will not provide you with the ability to build anything. Unless you redeploy it first. Though, funnily, the captured non-aligned Construction Yard is still seen as a Construction Yard, so will still provide a Production Boost Multiplier. This happens when you have a Construction Yard placed on a Custom Map that has a Faction of Neutral (not Goodguy or Badguy).
Custom Maps:
Goodguy is Allied or GDI
Badguy is Soviet or NOD
Neutral is not aligned and therefore a Construction Yard can't inherit Construction Options.
I originally noticed this particular behaviour on Tiberian Sun Custom Maps, so I am making an assumption that this behaviour originally came from the prior titles.