To check if it's actually a performance model, open the screen that shows the VIN and mileage and such, and look for a red line under the "Dual Motor" wording. If the red line under "Dual Motor" is present, it's a Performance, otherwise it's not.
I would also ask Tesla to confirm it by VIN, I think they would do that for you. Just in case someone's messed with the software to confuse it into saying it's a Performance when it's not officially. (Highly unlikely, but anything's possible and similar stuff has been done at least to Model S.) Any badge on the back of the car means nothing of course, I can stick on a Ferrari horse but that doesn't make it so.
Beyond verifying it's a Performance, do you want or care about the "Performance Upgrade Pack" (PUP) features? Note these features are standard now on 2021+ but used to be an optional line item. They are bigger (and better) brakes, bigger (but not better) wheels,
slightly lower (but not better) suspension, and I believe metal-covered pedals are part of it as well. Performance cars without this are often called "Stealth Performance" around here, or "P3D-" . Neither is an official Tesla term as far as I know. If you don't care for any of the PUP upgrades then feel free to buy a Stealth Performance, but make sure you're paying less for it than an equivalent car with the PUP. Functionally the upgraded brakes are the only true improvement in the PUP. The rest is just aesthetics and in fact the 20" cast PUP wheels are effectively a downgrade because they entail using a very thin sidewall that offers little protection from pinch flats or wheel damage, which is common with big Tesla OE cast wheels with rubberband tires.
What else to look for...consensus is that on average, build quality has steadily improved since the earliest cars. Also often newer revision parts. Something to keep in mind, though of course some older cars have had many issues addressed and parts replaced with newer revisions, making them better than they were when new (in some ways).
Starting in 2021, there was a bunch of small feature improvements made to all Model 3s (not just M3P). You can find out more with searching, but the standout ones to me are the heat pump, mildly revised and nicer interior (not hugely different but it's a bit nicer in a couple ways), and wind noise reduction from double-pane front windows & maybe other changes. It's still not luxury quiet but folks who've driven/ridden in older and newer ones back-to-back say the noise reduction is noticeable. Oh, and "blackout" exterior trim instead of Chrome. There are _no_ performance benefits to 2021 though (at least for M3P).
Some older M3P were sold with free unlimited supercharging (FUSC) (no longer available), however as I understand it that WILL NOT transfer to a new owner, FUSC was never transferable on the Model 3. So if someone's advertising that, they're basically lying (knowingly or not) because you won't get it. However I believe if it has free unlimited premium connectivity (FUPC) that MIGHT transfer, I haven't looked into that closely.
Many used Tesla buyers like to check the computer's estimate of the full battery capacity, to try to avoid buying a battery that has degraded more than typical for the age and mileage. Note both age and mileage and specific use affect degradation...every a very low miles older car will have battery degradation. There's lots to read on this, do some searching to find out more. It's important to note that ultimately the car is making an educated guess about the battery's remaining capacity, sometimes a car will think more capacity has been lost than actually has.
I actually don't know much about what to look for to verify the mechanical condition of a used M3P. Certainly a test drive would seem wise as with any used car. Normally I'd suggest getting a used ICE car inspected by an independent mechanic familiar with the car, but in this market sellers might just walk away if you ask for that, and for whatever reasons (some good some not) I think that's less typical for used Tesla buyers. Buying one that's still under fully warranty coverage is probably a good idea if your budget allows, since many/most are still under full warranty at this point. If you're buying direct from the owner, then pay attention to what they say about how they used the car, and try to get an overall impression of whether they might have abused their car (but don't be too rude in any questioning or probably you won't get straight answers).
Minor note: If a Tesla has no trim badge (e.g. the "dual motor" w/ red underline badge for M3P) it doesn't always mean the owner took off the badge, Tesla has delivered many cars without such badges over the years (not just Model 3). Tesla gave us the badge free ("goodwill") for our 2013 S P85 after we bought it lightly used many years ago from a private seller, but these days I would guess you'd have to pay a few bucks for an M3P Dual Motor badge if you wanted one and you buy a used car without it.
