they were pulled and told the foglights they had mounted maybe "DOT" approved, but weren't State Police approved.
There's no such thing as "DOT approved" fog lamps (nor headlamps, nor anything else). The US regulations do not work on an approval basis, they work on a self-certification basis: the manufacturer states/promises/certifies that the regulated equipment complies with all applicable provisions of all applicable regulations, No pre-approval by DOT is necessary or available; it simply doesn't exist. On a headlamp, the certification is made by marking the lens "DOT". There is no way to certify a fog lamp, because fog lamps are not Federally regulated. There is also no such thing as "SAE approved" lights; SAE is not a regulatory agency and does not approve anything. The state police do not have authority to approve or disapprove fog lamps. The regulation of auxiliary lighting on cars as to type, mounting, hookup and
usage is a matter of state law. The cops can enforce state law, and that's all they can do. Of course, that depends on the cop correctly understanding the applicable state law; if she doesn't, then the recipient of the ticket can fight the ticket on the basis of what the law actually says.
States are not permitted to prohibit Federally-regulated equipment that is compliant with the Federal regulations, nor are they permitted to require equipment that is not required by the Federal standard. No state can ban HID headlamps, for example, or require that all vehicles have daytime running lights. However, if a state wanted to pass a law that all vehicles not equipped with DRLs must use their low beam headlamps at all times during the day, that would be kosher. Likewise, states are free to pass (and enforce) laws about fog lamp usage that are as restrictive as they want, up to and including banning their use altogether.
In the early days of HID headlamps there were some cops pulling people over whose cars had fully compliant headlamp systems, and that was a matter of education, but usually when people get pulled over for improper lighting, it's deserved. The lamps are being misused (in dry weather, in daytime, etc.), the lamps are improperly aimed (glare), the owner has foolishly installed "driving" lamps (auxiliary high beams) and is using them in traffic, the owner is using the auxiliary lamps instead of the headlamps after dark, the lamps are producing light of the wrong color, etc.