Learn French: J'avoue que j'ai perdu sa confiance. - I confess that I have lost his trust.

In French, possessive adjectives match the gender of the thing being owned, not the person who owns it. The word confiance is a feminine noun, so it takes the feminine adjective sa.

Since sa matches the feminine word confiance, how do we know it means 'his' trust and not 'her' trust?

You can't tell just from looking at this one sentence! Because sa simply agrees with the noun it is attached to, sa confiance can mean 'his trust' or 'her trust'. You have to rely on the wider conversation to know who the owner is.

This lesson is about indirect object pronouns (lui and leur). Could we use lui here to say we lost trust 'to him'?

No, we can't. It is tempting to try and use lui here, but the verb perdre (to lose) does not use an indirect object pronoun to show whose trust was lost. Instead, you just state what was lost as a direct object using a possessive adjective: sa confiance (his trust).

Can I drop the word que like we often drop 'that' in English?

No, you cannot. In English, it is perfectly fine to say 'I confess I lost his trust.' However, in French, the conjunction que (that) is strictly required to connect the two parts of the sentence and can never be left out.