The continuum hypothesis is more like Euclid's parallel postulate than a Gödel sentence - assuming ZFC consistent there are models with CH true and CH false (the cardinality of the continuum doesn't have to be the first uncountable cardinal).

Everything gets qualified with "assuming ZFC consistent" or "assuming Peano consistent" because any inconsistent theory proves any statement. More of a proof technicality than anything too profound.

There is a construction of a model of Peano arithmetic, so it is consistent, as long as you accept the system used in the proof: https://en.wikipedia.org/wiki/Gentzen%27s_consistency_proof

Not sure if this sheds light on the parent commentator's question ... the terminology can be quite tricky.