It never was verification of US citizenship. You have never needed to be a US citizen to get a drivers license in any state as far as I know.
California required a birth certificate when I first applied for my license in 1973. The same was true for my middle son in 2000. The has been a perennial effort by Gil Cedillo to grant licenses to illegal aliens in California. Thus far it has been rejected. I've never seen a driver's license issued to a resident alien, so I don't know if they differ from ones issued to citizens.
The effort to require a government issued photo ID when voting has also raised the issue of whether a driver's license or state issued ID should come with verification of citizenship. The Real ID effort was supposed to address that issue, but it is getting rejected as an unfunded mandate by many states.