Would having more than one of these facilities increase the chances of getting extra abilities?
What about the healing bonus? Would two of those stack?
In XCOM 2, you can only construct one copy of most facilities, including the Advanced Warfare Center. The only facilities which can be built multiple times are Resistance Comms and Power Relays.
Regardless of whether or not you can build more than one AWC (you can't), you don't actually need to increase your chances, because the chance of a bonus skill is guaranteed*.
The way the game works, when the AWC is built (for existing soldiers) or when a rookie promotes to squaddie, a single random bonus perk is assigned, as is a rank (from Corporal through Colonel). Whenever the soldier reaches that level, they get the bonus perk.
* The exception happens in that the AWC only checks for exactly that rank, so if, by the time you complete the AWC, one of your soldiers is a Captain, and their bonus ability rolls at the Sergeant rank, they will never unlock it, because the AWC only checks their current rank each time they promote. (There is a mod that fixes this, however)