I ask this because Ferdinand Magellan's translator Enrique of Malacca spoke the Malay language, which also happened to be the language spoken in Cebu and Mindanao when they visited in 1521.
My family is from the Philippines, and I speak Tagalog. Tagalog has a low level of mutual intelligibility with other dialects of Luzon (e.g. Ilocano, Kapampangan ), and even less with dialects of other Philippine islands (e.g. Ilonggo, Cebuano ). While Philippine languages are still Malay languages, even when not counting the European influences, they have very little similarity with present-day Bahasa Malaysia and Bahasa Indonesia.
How did Malay languages diverge so much in less than 500 years? If so, do we have records on their evolution?
The diversion is much older than 500 years. Malay was spoken simply as a trade language in the Philippines at that time. It was not a language indigenous to the islands. Tagalog was (and is still) the primary language of Maynila and the surrounding area. Bisayan, Hiligaynon, and Tausug and other languages would have been the primary languages spoken in central and far southern Philippines (Sulu for Tausug), and still are. Basically, hundreds of languages were and are spoken all over the islands. Malay was like a kind of lingua franca for sailors and such.
I am not sure why you say there is little in common between Philippine languages and Indonesian languages. They are extremely closely related and the Malayo-Polynesian branch as a whole is noted for how similar the languages are from Madagascar to Rapa Nui. The big striking difference with Philippine languages is the Philippine Voice System, which fell out of use in other Malayo-Polynesian-speaking areas a very long time ago (before the languages we have today had even developed or spread to where they are now). The indigenous languages of Taiwan (which are Austronesian but not Malayo-Polynesian, generally speaking) do still have this voice system, so it is theorized to be a trait of proto-Austronesian. In any case, this very different, very morphologically complex grammar does indeed set Philippine languages apart from other Malayo-Polynesian languages, but it’s a very very old difference, far deeper than 500 years.
The languages of Indonesia and Malaysia have had extensive influence from European languages too. But that’s expected after 400 years of colonialism. Overall, borrowed words (which is basically the extent of it) still conform to the indigenous grammar and pronunciation. The loans become indigenized. I wouldn’t call loanwords a fundamental difference, personally.