What does "Battles Won" refer to?
This designated how many battles this Pokémon has won. In other words how many times it has successfully defended an attacker.
For further clarification, based on personal experience, I have only seen it pop up for Pokémon that took down my last Pokémon in my party when I was attacking a gym.
A few sources that back this:
Pokémon also display a "Battles Won: X" number that shows how many times that specific Pokémon has stopped an attacker -source
For each Battle Won, it means that your Pokemon was responsible for them losing the fight. That means their final attacking Pokemon was defeated by your guy. -source
Update: a change to gyms may have changed how this works. According to this reddit post,
I'm not sure if this is widely known, but the "Battles Won" that shows up on the pre-battle splash screen is not how many times that attacker was the victorious Pokemon when you defeated another Pokemon. It is actually how many times that attacker was part of a team that defeated a defending Pokemon. I noticed this because a Machamp I just caught from a raid had 15 victories this morning and I was pretty sure I hadn't used it that many times already. The 15 victories, though, referred to the 5 Pokemon I had beaten in a gym 3 times each last night even though I didn't use Machamp for each individual fight.
however, this is the only source I can find noting this change, and have not been able to confirm this myself.
As of the gym rework a few months ago, the Battles Won stat no longer refers to the number of attacking teams a Pokémon has stopped (by defeating the sixth Pokémon).
Prior to the update, Pokémon would rarely get into the double digits for their Battles Won stat due to the fact that trainers rarely let their teams be wiped out completely (to save face or to save revives), however, they now very easily reach triple-digit scores. This is because the Battles Won stat now refers to the number of defending Pokémon defeated while a specific Pokémon was in the attacking party. If you take down a full gym from maximum CP (defeat all Pokémon 4 times), then all Pokémon on your team will have their Battles Won stat raised by 24.
From this Reddit thread:
I'm not sure if this is widely known, but the "Battles Won" that shows up on the pre-battle splash screen is not how many times that attacker was the victorious Pokemon when you defeated another Pokemon. It is actually how many times that attacker was part of a team that defeated a defending Pokemon. I noticed this because a Machamp I just caught from a raid had 15 victories this morning and I was pretty sure I hadn't used it that many times already. The 15 victories, though, referred to the 5 Pokemon I had beaten in a gym 3 times each last night even though I didn't use Machamp for each individual fight.
I have put in a Pokémon and had a friend attack the gym and lose on purpose to see if the number goes up and it doesn't. So I don't think it has to do with defenses wins. Also have used a single Pokémon to take over a gym and then placed the same Pokémon in the gym and it hasn't changed. I'm sure it used to work when the game first came out but most likely no longer works just like the paw prints.