Learn Portuguese (Portugal): Nós queremos manter a amizade da nossa juventude. - We want to keep the friendship of our youth.

Because the action is directed at the noun a amizade (the friendship), not at each other. If you wanted to say 'We want to keep each other as friends' using a reciprocal pronoun, you could say Nós queremos manter-nos amigos.

Why does the sentence use da nossa instead of just de nossa?

In European Portuguese, possessive words like nossa are almost always preceded by a definite article. Here, the preposition de (of) combines with the feminine article a (required by nossa juventude) to make the contraction da.

Can I translate 'keep' using ficar com instead of manter here?

It would sound unnatural. You use ficar com when you keep or hold onto a physical object. For maintaining an abstract thing like a relationship, a habit, or a promise, manter is the correct verb.

Is the subject pronoun Nós mandatory here?

No, it is completely optional. The verb form queremos only ever goes with 'we', so you can just start the sentence with Queremos unless you want to emphasize who is doing the wanting.