Nobody is using Patriot to intercept Shaheds. PAC-2/3 are intended for ballistic missiles and fighter jets. In Ukraine, low-flying drones are primarily countered by FPV, MANPADS, drone-hunting aircraft and truck, and EW before point defense SAM and AA gun.
No one should use a Patriot to shoot down a drone, but they certainly have done so.
There are some pretty solid reasons why Ukraine offered consultation - unlike the US, they have actual combat experience against hordes of Shaheds. The US walked into a mess: they overestimated their ability to take out Iranian systems, and underestimated how hard it is to defend against them.
It's embarrassing. Now, the US seems to be looking for some way out, while preserving some dignity and pretending that they "won".
I did a quick search and could only found a missed missile having self destruction. Shaheds have a very distinct noise it shouldn't be too hard to id. While I don't doubt Americans had used patriot on drones, I believe it is because they were(and still are) unprepared and they panicked.
Actually they are using everything they have to combat these cheap drones. That includes Patriot and THAAD systems as well. Specially UAE, which got struck with more drones than Israel. That is how Iran was able to take out a THAAD radar, because it was deployed so close to them.
You can’t really deduce THAAD was used against drones based on this. As the number of US assets in the Middle East increases, it is logical to deploy more THAAD to protect them from Iranian ballistic missiles. Israel intercepted a few[0] at the beginning of the current conflict, so it is common sense to presume there are still some remaining.