Use of possessive adjectives in English

Solution 1:

The current terminology for words like my is ‘possessive determiner’, rather than ‘possessive adjective’.

I can’t speak for Spanish, but the French possessive determiners (mon, ma, mes, ton, ta, tes, son, sa, ses, notre, nos, votre, vos, leur, leurs) are frequently used. Reflexive constructions like Me duele la cabeza are also found in French, but that is really rather incidental to the use of possessive determiners. Se me cayeron los anteojos is presumably a passive construction, but I’m not sure it has much bearing on the use of possessive determiners either.

Old English had a set of possessive determiners in the first, second and third persons, and singular, dual and plural numbers, and the third person form was, additionally, inflected for the gender (masculine, feminine or neuter).

Romance languages use more impersonal constructions than English because Latin also did so, but that is really a separate issue from the use of possessive determiners.

Solution 2:

Your observation of the reflexive might actually be use of Middle Voice. Your body part is taking on a role in the sentence that is neither active nor passive.

In French (and I would also assume in Spanish), there are two forms of the verb "to break."

Je casse ma bicyclette. ("I break my bicycle.")

Je se casse ma jambe. ("I break my leg," Literally I my-leg-breaks-itself)

And when I shave, the one being shaved is between active and passive.

Je me rase. ("I shave," Literally I shave myself)

(And my apologies to French speakers everywhere when I break their language.)

Note the use of the middle voice ("se casse") when I break my leg. Middle voice is used because the breaking is between me actively breaking my leg and my leg having passively been broken.

You may be seeing this use of the reflexive / middle voice in romance languages because English does not have a middle voice.