Can an illegal alien use a matricula consular id to marry someone of the same sex?

No

Yes, but it is better to use your Mexican Passport. If he doesn't have one, he should get one for this and other purposes. About a dozen US states accept the matricula consular as valid photo ID, including Washington, which also permits same-sex marriages. Even outside of these states, you could probably get the marriage license using the matricula consular alone, but it would be better if you could bring a birth certificate. If you are thinking that you need the matricula consular to prove residence or legal address ... don't worry about it. You don't need to prove where you live to get a marriage license. Foreigners are allowed to get married in the U.S. You don't need to try to appear like a citizen.