I don't know about the why's and wherefore's of Drakeford's plan and whether it is right or proper etc.
I've been to Wales countless times - my best mate lived near Bangor, mother in law lived in mid Wales, been across the bridge countless times to go mountain biking. I have rarely, if ever, encountered anti-English sentiment. The apocryphal lines about people in pubs switching to Welsh when a stranger walks in..... never happens.
Enforcement can and has been done by the Police. ANPR will pick up the registered owner immediately. The likelihood being that if it's not a locally registered car, you may get pulled. Company cars (which are a minority) have the registered owner as the lessor but if they're on finance, then it will be the registered owner.
Now are they going to pull every car? In the same way that they will ask every rail passenger? No. As
@Bletchleyite says...
It is enforceable in the same manner as PFs on a railway with open stations. You do random checks and hit people with a hefty penalty "pour encourager les autres".
The potential to get stopped and fined is the strategy that will be employed, and yes, as
@bramling suggests, they will be targeting certain honeypot locations. So if you drive to Ogwen Cottage on the A5, then there's a chance you'll get a friendly welcome from the local constabulary. However, drive over the tops from Oswestry on the myriad of roads towards Llangollen and go walking round there, and you'll be unlucky.