In many parts of the States you do, but the strictness of the test varies and the standards are grandfathered.
So, for example, in California (one of the strictest) you must pass a visual inspection, a tailpipe sniffer, and an engine computer scan. There are also many rules about acceptable engine modifications. In other states you may only get an engine computer scan.
As for standards, cars are (rather reasonably) held to the standard they were made to, not the standard of today, and modern cars really are dramatically less polluting than cars from several decades earlier.
Even in California, there are exemptions for: "hybrids, motorcycles, trailers, or gasoline powered vehicles 1975 and older."
...which seems shortsighted. Except for trailers, all of those can have mechanical problems that cause it to pollute far more than they should. My daily driver for a long time was from 1974 and I was never sure why older cars were exempt. If it's a hobby car with historic plates, sure, but if you're driving it every day you should have to follow the same rules as everybody else. In my defense, my old car got 68 mpg and was well fussed-over.
Motorcycles trail cars in emissions improvements & regulations, but they follow nonetheless. Things like electronic fuel injection was not practical on motorcycles until more recent times.
Part of the rationale with the 1975 exemption is that vehicles 1975 and older represent a very small portion of the number of cars on the road, and are driven far fewer miles than average. That beautiful 1966 Corvette didn't last fifty years by being driven every day.
So, for example, in California (one of the strictest) you must pass a visual inspection, a tailpipe sniffer, and an engine computer scan. There are also many rules about acceptable engine modifications. In other states you may only get an engine computer scan.
As for standards, cars are (rather reasonably) held to the standard they were made to, not the standard of today, and modern cars really are dramatically less polluting than cars from several decades earlier.