Since "we" = subject, and "our sons" is the direct object, we've got a reflexive sentence (where subject and 'possessor' are the same persons). I think "we love our sons" would 'normally' be: Fīliōs amāmus. So, whoever is saying this sentence is laying great emphasis on the fact that We love our own sons. I think Duolingo ought to accept "our own" as a translation for nostrōs , in this context (where "we" are also the subject). (edited to add macrons.)
You are no doubt correct about the latin part (use of noster), but you are mistaken about it being a reflexive sentence.
We love our sons.
'Our' is not the object. Only 'sons' is the object receiving the love. 'Our' merely serves as an adjective telling us 'which sons'.
So the sentence is not reflexive, as 'we' are very separate entities from our 'sons'.
But the biggest and most difinitive clue that this sentence is not reflexive is the lack of a 'being' verb, or form of 'esse' in latin.
However, 'I am my own grandpa' would certainly be reflexive.
It's true that "We see ourselves," or "We hurt ourselves," are obviously reflexive sentences, because in them, the subject ("we") reappears as an object ("ourselves," either accusative or dative, depending on the verb).
I think of the sentence, "We love our sons," as REFLEXIVE because the subject 'reappears' as the POSSESSOR: that's the meaning of "our," after all. (If "You love our sons" is the sentence, it's not reflexive: subject (YOU) and possessor (WE) are not the same group of people.)
If you study the 3rd person possessives in Latin--the REFLEXIVE possessive adjective suus, sua, suum (his/her/its/their OWN), versus the non-reflexive genitive-case pronouns eius (his/her/its) and eōrum/eārum (their)--you'll see that it can be useful to consider the "possessor" function as reflexive (as well as the obvious object functions).
If you know the difference between "He carries his (own) trunk" and "I carry his trunk," this will make sense:
Cistam suam fert . reflexive possessive adj, so possessor and subject are the same. "He (or She) carries HIS (HER) OWN trunk."
Cistam eius ferō . "I carry HIS (or HER, or ITS) trunk."
(Use of "being" verbs has nothing to do with being reflexive, by the way.)
I think it might just be a matter of terminology--that you don't want to see as "reflexive" the 'possessor' function. The 3rd person possessive situation in Latin, however, hinges on precisely this.