TLDR: "Compact" is a legal fiction to get the right to build a building. It's pretty much useless.
The "compact car parking space" is about entitlements aka the right to build, which is decided at the local level. Cities often require a minimum number of parking spaces for a development. This then requires defining a minimum space size and aisle size. And builders pressure cities to allow a percentage of "compact" spots with smaller space and sometimes aisle size. An example of code in Pinole, California (exurban San Francisco, and the kind of place you need a car to get by).
A. Up to twenty-five percent (25%) of the required number of parking spaces may be sized for compact cars.
B. Compact car parking spaces shall be at least eight (8) feet in width and sixteen (16) feet in length, and shall be clearly marked, “COMPACT CARS ONLY,” “COMPACT,” or “C.”
C. Compact car spaces shall be distributed throughout the parking lot.
D. Where a section of the parking lot is restricted to compact parking with an angle of 90 degrees, the aisle width may be reduced from the standard twenty-three (23) feet to twenty- one (21) feet. Such compact sections should be located so as to minimize the distance from the section to the appropriate building or activity.
Note absence of state-level standard, even in California. The takeaway here is this a local matter and will be subject to local enforcement if any.
Police do not (by default) have standing to enforce on private property. Some departments are willing to sign contracts with (larger) parking lot owners to enforce on their property - this will be the case at a large mall, typically. Such consent is a double-edged sword, and smaller entities will think twice about doing it.
Even when they do, "Is a car a compact car?" is too vague and slippery a question, because again it is defined on a local level. This is a vexation to people who find Tacomas and Dodge Chargers parked in the compact spots.
Your best bet is to appeal to the private property owner. The host makes the rules, and they can make ANY rule they want. And yeah, they can enforce it with towing, as long as the parker has a reasonable opportunity to know those rules, and the rule is sane enough that a judge doesn't call it a predatory tow. Penalty tows HURT - they are many times the cost of regular old rescue tows. They can say "your vehicle may only park in the Designated Loser Spot for the next month as punishment", or "pay a $60 fine or don't bring the vehicle back here" or just "banned for a month", with towing-on-sight being the penalty for noncompliance. So it really depends on the righteousness of the property owner.
Remember though - the property owner only created those spots to get a license to build. So now that they have their entitlement, they're not emotionally invested in the principle.