Among Malays in Malaysia, they identified themselves into 3 groups, i.e. anak jati (the real ethnic Malay), anak dagang (other ethnic groups like Javanese, Buginese, Minangkabau, etc that assimilated into Malay) and peranakan/jawi pekan (those of Arab or Indian ancestry). Some mixed mamaks can be considered under the 3rd group where you can find a lot of them in Penang/Kedah. Most of anak jatis concentrate along the northern states while the anak dagangs along western coastal states like Selangor, Negeri Sembilan and Johor.

However, given that the constitution in Malaysia and the definition of Malay is political, they consider themselves all Malay belonging to the greater Malay race of Nusantara/Malay archipelago. In Indonesia, it's a different story as each ethnic group have their own languages, written language and culture although there are some overlaps due to influences here and there. Most Indonesians are ethnic Javanese (about 100 million or more) with their own proud culture and history that predates Melaka so I don't think they would call themselves Malay as that would refer to the much smaller group in Indonesia.