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That's actually a matter of some contraversiality.

You can't actually just replace "my" in a sentence with an adjective and have it come out sounding natural. You can say "this is my house" but you can't say "this is big house". You're missing a determiner, not an adjective.

Possessive determiners are determiners which express possession. Some traditional grammars of English refer to them as possessive adjectives, though they do not have the same syntactic distribution as bona fide adjectives.[1]

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The words my, your, etc. are sometimes classified, along with mine, yours etc., as possessive pronouns[3][4] or genitive pronouns, since they are the possessive (or genitive) forms of the ordinary personal pronouns I, you etc. However, unlike most other pronouns, they do not behave grammatically as stand-alone nouns but instead qualify another noun, as in my book (contrasted with that's mine, for example, in which mine substitutes for a complete noun phrase such as my book). For that reason, other authors restrict the term "possessive pronoun" to the group of words mine, yours etc., which replaces directly a noun or noun phrase.[5][6] — Wikipedia, Possessive determiner

This is further complicated by the fact that some words are sometimes true pronouns, and sometimes possessive determiners (his, her, its). In this way, it is difficult to fully separate the role of possessive determiner from the role of pronoun.

But thank you for making me research it a bit more.