Yes, adobo is from Spain and it spread to Spanish colonies including Mexico. Every colony that adopted the Spanish adobo developed its own peculiar style over time. But, no, Filipino adobo is not the Spanish adobo that that became integrated in the cuisines of Spanish colonies.
Mexican adobo is a sauce made with chilies, herbs, vinegar, salt and a bit of sugar to balance the acidity. A stew cooked with adobo is adobado or adobada. It is a delicious stew, not the same as Filipino adobo but just as comforting and satisfying.
Mexican adobo is, naturally, made with chilies that grow in the region — jalapeño, in particular or its smoked and dried form, chipotle.
Because I live in the Philippines where jalapeño is not cultivated for commercial purposes, when I made adobo sauce to cook pork adobado, I substituted the locally grown bird’s eye chilies. They aren’t as fleshy but they are hotter. To add even more depth of flavor, I pan-roasted them until charred to add some smoky tones to my sauce. I also used a food processor to make chopping easier although I merely pulsed the ingredients so that visible bits remained. I do prefer chunky sauces over ultra smooth ones. »