The "Matricula Consular" is an ID card issued by the Mexican government to their citizens. It's actually an old form of ID that wasn't often used until recently, when that government decided to give it to the millions of illegal aliens from Mexico who live in the U.S. in order to make it easier for them to send their money home via remittances. That government then began a campaign to get banks, municipalities and others to accept their cards as ID. Make no mistake about it: these cards are almost entirely simply "IDs for illegal aliens". Legal immigrants have no use for the card since they have legally-obtained identification which they're required to carry.
The Mexican government - and their U.S. enablers - are generally coy about the use of the cards and will only say that they're given out without checking someone's immigration status. However, one of their consular officials did admit what their real purpose was.