health care checks. Hotel check in. Employment? maybe. Background Checks? doesn't matter.
It actually does matter, because there is sometimes a law governing the documents that may be shown for a given purpose.
For example, the I-9 form, for verifying someone's eligibility to accept employment in the US, has a well defined lists of documents that an employer must accept, and the passport card is one of those documents. A similar situation exists for Transportation Security Administration screening of air passengers.
On the other hand, laws concerning proof of age for buying various products will vary from state to state, and retailers may or may not be required to accept any particular document. In the case of alcohol sales in North Carolina, for example, there is a brochure that lists "acceptable forms of identification" on page 17 and explicitly says that "passports may be in the booklet or card form." But that does not seem to create a legal requirement for the retailer to accept passport cards, because page 19 outlines the retailer's right to refuse, saying among other things that "there is no legal recourse by a customer who you have refused a sale."
US passport law (22 USC Chapter 4 and 22 CFR parts 51 and 53) doesn't have anything to say about the passport's or passport card's role as an identification document; it speaks only of the more specific role as a travel document.
So the general answer to your question, appears to be no. There is no law generally requiring people to accept a passport card if they also accept passports or driver's licenses.
But in most specific instances, there may be a general requirement such as "government-issued identification" that includes passport cards in addition to passports and driver's licenses, or there may be a list that explicitly includes passport cards along with driver's licenses and passports.