I just saw a youtube video of the 6.5mm CBJ cartridge which travels through 7mm armored plating.

Which handgun calibres and ammunition have armor piercing capability? Or: (how) can regular ammunition be converted to armor piercing ammunition?

The Tokarev is generally known for this capability, although mostly from the carbine.

9mmP with THV is said to be AP, in which case the 357Mag should be AP as well.

Does anyone have better knowledge or experience?

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I think the AP ability would be directly reliant on bullet material and bullet construction.
Original vest penetrating bullets were simply non deforming pointed bullets if my understanding is correct. Cop Killers.
Armour piercing could/would be made using a soft deforming outer which reduces ricochet likelihood and a solid high density inner core of a reduced diameter - stuff like depleted uranium or tungsten.
This was on the net :  Depleted Uranium (DU) and Tungsten are both high-density heavy metals used for armor-piercing projectiles, with DU generally offering superior penetration due to its "self-sharpening" property and pyrophoric (incendiary) effect, while Tungsten is a non-radioactive, safer alternative, albeit sometimes less effective against advanced armour. 

https://youtube.com/shorts/vXr8p2LsKYI?si=6Q9xVA-9eSDhYhSE

This round is a sabot containing a thinner projectile. It is similar to the Tokarev design ie a neck down cartridge with very high velocity bullet.

Lead, I think.

I have observed rifles shooting exposed lead tip bullets make round holes in steel plate at 100 meters, where monos and full jacketed just mushroom.

Similar effect should exist for handguns, proportional to the velocity decrease.

Lead, I think.
I have observed rifles shooting exposed lead tip bullets make round holes in steel plate at 100 meters, where monos and full jacketed just mushroom.
Similar effect should exist for handguns, proportional to the velocity decrease.

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I do believe other factors -velocity, bullet weight etc were the cause of what u saw. I doubt the lead would have out performed the copper- brass if same weight, velocity etc were used on the same plates.