What is this phrase, "I hope we catch up," called when "catch" may be confused with what's done with a baseball?

This is called an idiom. An idiom is a phrase that has a figurative meaning that differs from the literal definitions of the individual words.

Here is a list of english idioms. Several variations of "catch up" are on the list.

There could possibly be an occlusion of "...to [object in question]..." in the phrase "catch up..." In that case, it would make sense to use "up to", as it would be demonstrative of a spatial orientation between two objects (namely, the speaker in question in relation to the person whom he/she/they is/are trying to catch [up to]). Therefore, using "catch" is practically the same when applied to people or spherical objects, but adding "up [to]" adds the notion of two objects approaching one another.

Catch up is a phrasal verb and so the two words should not be separated in this context.

Therefore, we are not dealing with the word catch as in catch the ball,

but rather catch up as in If you run fast enough, you might catch up to the others.

Here, catch up means to close a gap between yourself and someone who is ahead of you.

When we use it in this sense, we are catching up to the time that has passed; Since last I saw you, many things could have happened to me so my 'life story', in a way, has moved on from what you know. When you meet me and I tell you what has been going on, I catch you up, meaning 'bring you up-to-date'.

The important thing then to realise is that catch and catch up are not the same.