Founded in 2004, MAPI is managed jointly by the Municipality of Montevideo and private partners, reflecting a shared cultural vision. The museum's initial collection was donated by collector Matteo Goretti under an agreement with then-Mayor Mariano Arana. Since then, MAPI has expanded its holdings through donations from institutions such as the University of the Republic and the National Library of Uruguay, as well as embassies and private collectors is a cultural institution with mixed management under the Department of Culture of the Municipality of Montevideo, in which the parties contribute values, heritage and resources from a shared vision.

It was created by Decree 31.383 of the Departmental Council of Montevideo and received its first collection through a gratuitous bailment agreement signed between the then Mayor of Montevideo, the architect Mariano Arana, and the collector Matteo Goretti, who donated part of his important ethnographic and archaeological collection for a period of 20 years. More recently, MAPI's holdings have been enriched by twelve other collections donated or loaned by public institutions - the Faculty of Architecture of the University of the Republic, the National Library of Uruguay, among others - embassies and private collectors.