Learn Polish: Trener uczy młodych ludzi cierpliwości. - The coach teaches young people patience.

This is the specific grammar rule for the verbs uczyć (to teach) and nauczyć (to teach successfully). The person receiving the teaching is the direct object and takes the accusative case (młodych ludzi), while the subject or skill being taught takes the genitive case (cierpliwości).

Since masculine-personal nouns look the same in accusative and genitive, is młodych ludzi actually accusative here?

Yes, grammatically it is accusative. While the forms for the accusative and genitive are identical for masculine-personal (virile) plural nouns, the structure of uczyć dictates that the person being taught acts as the direct object in the accusative case.

How do we form the genitive for a word like cierpliwość?

Cierpliwość is a feminine noun ending in a consonant (-ść). Most abstract feminine nouns with this ending simply take an -i at the end in the genitive case, making it cierpliwości.

Why use the imperfective uczy instead of the perfective nauczy?

Uczy emphasizes the ongoing process or routine: the coach is currently in the process of teaching them patience. The perfective nauczy would mean the coach will successfully teach them, focusing entirely on the completed future result.