Powermatch

No. Only a 3rd party app can do such a match, like Zwift or TrainerRoad.

Maybe it works with your Core too in the Wahoo own app: www.smartbiketrainers.com/how-to-enable-the-wahoo-kickr-snap-powermatch-feature-and-why-you-should-do-it-5726  bottom of the page. 

Today I trained in ERG mode. I had the power set to 165W and the meter was still showing something around 180W. Shouldn't it be like that?

It's "normal". As Blake said there is no match in Edges so You will have difference on displayed power (from pedals) comparing to target (trainer). Did You calibrate both devices properly? What kind of pedals? And dual or single ones? Between 2 perfect devices (pm and trainer) there is difference too due to drivetrain loss, ok, it should be 2-3watts maximum with such a power. 

Assioma favero duo.

The trainer itself has a powermatch function, unfortunately it can only be turned on with the iOS application. I don't think anyone has iOS among my friends :(

The Assioma Duo should be accurate if zeroed/calibrated before every ride (and crank length setting is same in the Favero app and in Edge too, is it set?, i mean in  the Edge under sensor detail of the Assioma, a bad crank length setting in Edge overwrites the Favero's app setting and in pedals and can cause a few percent under or overmeasuring).  The Core must be calibrated weekly after a proper warmup, if it didnt happen, it can underread the power (but usually the Cores are overmeasuring it as i experienced it, and the Core's accuracy is much worse than top-end trainers or top-end bikepowermeters so usually there is/will be difference...) . And check Core's firmware too, is it the latest or not.

Check these , maybe the difference will be much less next time . (I hope so)

What kind of mobile do You have? Thr Zwift is running good on lots of mobile (even on 4-5 years old Samsung S6 too)

"The trainer itself has a powermatch function, unfortunately it can only be turned on with the iOS application. I don't think anyone has iOS among my friends :(" it works only in (IOS) Wahoo app then, so You should own an IOS device for workouts (and not just borrow one , this Wahoo setting is not working with other apps or head units just in Wahoo app)

Why doesn't the Edge 530 support powermatch? If ICQ applications can, why can't Garmin?
I think that's quite a useful feature!

which "ICQ" (???) app can do it? the apps what i wrote running on computers or on mobiles/tablets, these have much more resources (cpu, memory, battery), this powermatch function needs continously monitoring 2 power sources and their differences (aka offset) and adjust it very often, these devices (cycling head units or wristwatches, so neither the Wahoo or any other brand's head units cant do it and wont do it for years for sure) designed for (more) simple functions... 

or do You think for IQ app XERT? i think much better and reliable to use XERT on your mobile , let it control your trainer and doing match and your Edge only record your pedal's power in the FIT file. But it needs XERT subscription too... 

try the Zwift on your mobile, it has trial , You can test that the powermatch is working in mobile or not (i dont know it 100% sure, on computers, WIN and MAC, it works, i suppose it should work on portable devices too but never knows...) and if it works You subscribe , much better do workouts in Zwift than just watching that small display on Edge and its numbers, not boring, and the time goes much faster..  

Xert player