Are Mexican "albures" the equivalent to American "puns"?

Yes, in Mexico, albur means pun, so albures = puns. In Castilian Spanish, they say "juego de palabras" = "play on words".

There is no such thing as "Mexican" Spanish. Only if you are implying that in Spanish speaking countries the language is spoken differently, then, you are right.

"Mexican" Albures IS the equivalent to American "puns."

Albur (verb) is a figure of speech which consists of a deliberate confusion of similar words or phrases for rhetorical effect, whether humorous or serious. Mainly is a "word game," used most of the time sexually. Albures are used most of the time to disguise a "profanity" without actually saying it, or it changes the structure of a sentence by replacing or adding a word, taking a word out, and making jokes out of them.

i.e.:

q:¿Cuál es el pez que da leche ?

a: el PEZón

q: Which fish gives out milk?

a: the nipple

Pez: fish

PESon: nipple

Me chingaste cabezón:

RESPUESTA: Tu por pendejo yo por chingón

You fuc.ked me up good bighead/pighead:

Answer: Because you are fuc.k and I'm a *******