As a single-classed brawler,¹ you cannot take the Advanced Armor Training feat: while the brawler’s martial training allows them to meet the “fighter level 3rd” requirement on Advanced Armor Training, that feat also requires the armor training class feature—which brawler doesn’t offer.¹
However, the question remains valid for a multiclass fighter/brawler that does have armor training (i.e. at least fighter level 3rd). In that case, RAW, martial training does two things:
Makes your brawler levels count as fighter levels (and monk levels) for qualifying for feats.
Makes you count as a fighter and a monk for feats and magic items that have different effects based on whether or not you are a member of those classes.
That is, unfortunately, it: the first one, that actually involves your level, doesn’t change anything about effects, just qualifications, and the second, that does affect what things do for you, doesn’t change your level, it only changes the answer to that yes–no question of whether or not you’re one of those classes. RAW, a fighter/brawler only gets to use their actual fighter level to determine the effect of advanced armor training effects (whether gained through fighter levels or the feat).
However, both of the examples that are given for the second effect—a monk’s robe or Stunning Fist—don’t really care about the straight yes–no answer that the second effect offers. You need some number of monk levels for those things to have any different effect. That certainly implies that despite what was written for the second effect, it actually should also include your brawler level in those levels.
This contradiction, unfortunately, isn’t really resolvable. Under D&D 3.5e rules, the rules text is “primary” and the examples “secondary,” and so we’d go with the rules text, but although Pathfinder is based on D&D 3.5e, Paizo never actually copied those rules. Generally speaking, they were loathe to discuss the possibility of errors in their products, and when they did all we got was “ask your GM.”
Mind you, in this case, even in 3.5e where we have an “official” answer here, in my opinion anyway, it’s really dumb. I would happily allow you to include your brawler level for advanced armor training effects even if RAW clearly said otherwise. It seems better balance to me, and it seems to be in keeping with what the brawler is supposed to be. Given the nature of the examples, the rule was probably just poorly written. But anyway, they actually wrote the rule and the examples, so you should ask your GM what they think of the situation.
- I haven’t investigated the brawler’s archetypes to ensure this is always true. It’s true of the base brawler, anyway. If an archetype offers armor training, then the first paragraph doesn’t apply and you can take Advanced Armor Training as a feat, as discussed for a fighter/brawler in the rest of the answer.