How to tag a Smog Check Center only, in America?

Emission tests, which is what I presume you mean by a Smog Center, are a standard part of having a road-legal vehicle in many countries. In the UK they are included in the MoT test, and I know the Swiss standards are fairly stringent. Usually these are part of a suite of tests, so something like amenity=vehicle_test_centre (note OSM standard is to use British English) would seem appropriate.

Many car repair/service garages will also offer these tests, in which case shop=car_repair with vehicle_test_centre=yes, would enable garages which also do tests to be separated out. Some in fact specialise in performing the test, with car servicing as a sideline, in which case one could use car_repair=yes. (AFAIK we dont distinguish between places which service cars & those which deal with accident damage repair).

If the facility only carries out one kind of test then use either vehicle_test_centre=emission or perhaps 'namespace' it with vehicle_test_centre:type=emission.

None of these suggested tags are in use, although there are several well-established precedents in similar areas. My main point would be to try and use a tag which works across different countries & jurisdictions: firstly it is likely to get better take-up, and secondly it will make life easier for anyone who wants to use it downstream