Jose Ortega y Gasset was a relative of mine, and I asked this as a kid. What I was told (by my grandmother, his niece) is that it is sometimes done when there's a weirdness in the combination of the two last names that would make it ambiguous, such as someone who has a compound first name like Jose-Maria that makes it harder to tell where a first name/last name breaks, or in this case, where it results in a repeated sound (ga-ga) that feels weird to say. But I have not found any corroboration of this online.

It was a common practice in Spain until the XIX th century. E.g. Miguel de Cervantes y Saavedra (1547-1616), the author of the book "Don Quixote" used the same pattern (in modern texts is often referenced without the "y", as Miguel de Cervantes Saavedra). Literally, it means Miguel "from" (as "son of") Cervantes (father first surname) "and" Saavedra (mother first surname).

Currently that form is in disuse, so if being born nowdays it would be just José Ortega Gasset, although people could formally/informally use "José Ortega y Gasset", "José de Ortega y Gasset", or even "José Ortega i Gasset" (in case of being written in texts in Catalan, instead of Spanish, a form that is still in use in some Spanish regions having the Catalan/Valencian in addition to the Spanish).