So I'm just at the beginning of MM2, and I decided to buy my robber the Gambling skill because the manual states that it increases luck. However, I found no change to my luck statistic (it's base level is 20). Is his luck maxed out somehow or does the skill work in some other way?
Another character was able to boost one of their stats to 25 by equipping an item, so it doesn't seem like there's a problem with having stats higher than 20.

I looked it up on GameFaqs.com, and it says it should give +5 to Luck. It also says the max stat is 255, so a 20 luck shouldn't be maxed out.
Don't know why it didn't go up for you, unless your Robber didn't have the Gold on him to actually make the purchase. Maybe another party member has it all "gathered?" ...just trying to think outside the box.

So I loaded up the game again and now my robber sees the bonus. I think maybe it just didn't update until I saved the game.

The game keeps separate tracks of what your stats are, and what they should be. Resting normalizes them.

Yeah, the thing that confused me was that equipping an item that changes your stats takes effect immediately. Boosts from secondary skills, like you said, don't take effect until after you rest.