Yes. The "Indians" also weren't one big happy family either --- there are many different groups of them. A Tarahumara has little in common with a Mayan and so on. Both would be lumped together under the umbrella word "hispanic" even though they have a different culture and language and often don't speak Spanish. Vicente Fox himself has an Irish last name and his mother came from Spain -- not really what most people consider a typical Mexican --- but then most people will call people in the USA with Spanish blood especially in areas of the SW "Mexicans" even if they never once stepped foot in Mexico and if their families never lived there.
If you watch Mexican TV shows on the Spanish channel, I see lots of Whites. Many of them have non-Spanish last names, often Irish, English, German, Italian, Polish, Russian, etc.