How long did it take for Europeans to realize that the land they had discovered across the Atlantic was in fact not India? Surely it couldnt have been that long, in which case why did the name "Indian" or "Red Indian" stick for so long?
It's kind of odd when you think about it.
In addition to the great answers linked by /u/Georgy_K_Zhukov, I've also written about this for the Latin American context before, where this developed a bit differently from the US:
Spanish names for the Aztecs and vice versa (esp. point III)
For the region: How did the Spanish call the Americas before that name caught on? (las indias, nuevo mundo etc.)
A very short summary of those:
The term "indio" for native groups had first been used by the Spanish in the Caribbean, due to Columbus' famous mix-up involving his supposed discovery of "India". The mix up with Asia was cleared up in Europe at least through Magellan's circumnavigation of the world, finished in 1522. By then, however, Las Indias had already become the standard official Spanish term for their overseas possessions, and stayed in use until the 18th c. This was in contrast to other European powers like the English and French, who used terms like "America" or "New World" more regularly.
On to your main question: Due to the general use of indio for all native population groups, pre-colonial identifications were strongly simplified and homogenized, as in most other Spanish American regions. Already in the 2nd half of the 16th century casta terms including indio, but also espanhol and mestizo played an increasingly major role in colonial administration and the justice system (not just in Mexico). Spanish was the only admitted language before colonial courts, which further increased the role of the indio concept for Spaniards and native people alike.
The strongly discriminatory nature of the indio term was only officially recognized in the early 19th c. The newly independent Latin American nations would start using "indígena" (indigenous) instead to avoid those pitfalls - all the while continuing to exploit native groups. Today, indígena is still in use; other terms like pueblos originarios have come up; and some population groups (understandably) prefer to be known only by their own group name, so context is key here.
More of course can be said on this topic, but /u/snapshot52 has written on this before. I would point you to the entry he wrote on terminology for the /r/IndianCountry FAQ which you can find here, which is at least a good start.