That's how the IDs are used and most probably the motivation for their issuance. But just because the IDs are overwhelmingly used by unlawfully present aliens as a general-purpose indentification card does not make the issuance of the card illegal and does not take away the fact that there is a legitimate diplomatic reason for the issuance of such a card (even though that is not the actual motivation behind the country's issuance of the card).
The notion that accepting the card is a crime is simply false because the cards provide no information as to a holder's legal status in the United States. (They can be issued to legally-present Mexican nationals as well though, I imagine few use them).
I'm an accountant, not an attorney, but I would imagine that in the eyes of the law, presenting one of these cards creates the presumption of being illegal (probable cause to investigate your immigration status?)
Logically, legal Mexican residents have no use for the card, as they are eligible for US-issued identification.
Also - if the Mexican government wants these cards to be considered more "secure", and accepted in more places, they should issue them responsibly. First of all, by verifying a person's actual identity, and only issue them to people with provable legal status in the respective country (USA).
As an American (and its basically the same thing here in Canada), you cannot get a driver's licence, health card, passport, etc., without proving who you say you are.
Neither the Canadian or US governments issue such "consular cards" to its citizens living abroad, but they do register citizens abroad who wish that.
Though the quantum of proof is low (basically unverifiable), and no special ID card is issued, they do ask for your immigration status in the country of question, and "illegal" isn't a choice.