Some of the VMC is pretty straight forward when it comes to abilities vs. "levels" and some are not. Some magic items are pretty straight forward how they relate to abilities and some are not.
For instance:
Bravery: At 3rd level, he gains the bravery class feature as a fighter of his character level – 1.
Armor Training 1: At 7th level, he gains armor training 1.
Weapon Training 1: At 11th level, he gains weapon training 1.
Armor Training 2: At 15th level, he gains armor training 2.
Weapon Training 2: At 19th level, he gains weapon training 2.
Sash of the War Champion (Fighter level is 4 higher for Bravery and Armor Training class features)
Since Bravery has an "equivalent" Fighter level, it would be easy to add 4 to it to find out what the Bravery bonus would be, but Armor Training 1 does't have a level associated with it. Fighters get -1/+1 for ever 4 levels, so the magic item would give a fighter with Armor Training 1, -2 to ACP and +2 Max DEX (the equivalent of Armor Training 2). Would the Fighter VMC get -2/+2 for Armor Training 1 with the item or not gain anything since Armor Training 2 for the VMC isn't available until 15th level? (I think this is along the same lines as the Myrmidarch ability of the same name that "stacks" with Fighter levels, but doesn't have any Fighter levels). Would the Fighter VMC gain Armor Training 2 at level 11 with the item (4 levels sooner than without the item)?
Gloves of Dueling on the other hand (pun TOTALLY intended) gives anyone with the Weapon Training class feature +2 to whatever level they have it at. So a Fighter VMC that was 11th level would get +3 to hit and damage with the one group they choose. Item works just fine for the VMC (and any class/archetype that has the class feature). Not based on level.
Since both Bravery and Armor Training advance every 4 levels, would it break the world if the item was reworded to match the Gloves? "Anyone with the Bravery or Armor Training class features increases their bonus with each ability by 1". That would allow Myrmidarch, Dragoon, and other classes/archetypes that have Armor Training benefit from the item without having actual Fighter levels.
There are other combinations (Sorcerer and Robes of Arcane Heritage, Monk and Monk's Robes, Druid and Druid Vestments, etc. that might have similar wordings that don't work RAW but probably should RAI, it has just taken me most of the day (at work so not able to focus on it) to get these two examples down.
Would the Fighter VMC gain Armor Training 2 at level 11 with the item (4 levels sooner than without the item)?
No. Unless the ability stated that the character gains armor training as a fighter equal to his character level (or any other variant of character level -something), the ability is a static ability and does not change.