Is ammonium a strong or weak conjugate acid?
Both ammonia is a weak base and ammonium ion is a weak acid.  Many, even most, acid/base conjugate pairs are like that.  We should be using comparative instead of absolute adjectives in the rule about conjugate acid-base strengths:

A weaker acid has a stronger conjugate base, not necessarily a totally strong one.

A weaker base has a stronger conjugate acid, not necessarily a totally strong one.