What is the difference in meaning between “I think” and “I am thinking” in the sense of having an opinion?

Verbs of inner state, unlike most verbs, do not normally take the progressive in English.

So I think (I feel, I believe) are the ordinary unmarked forms.

When they are used with the progressive, this is “marked” – not the normal form – and generally emphasises that the inner activity is going on at this very moment.

So if somebody says “I’m thinking that … ” it says that they’re thinking that at that moment, with the implication that they might not think that at other times: perhaps they’ve just realised something; or they are working out what they think as they are saying it; or even perhaps they half expect somebody else to contradict them and change what they think.

Similarly in your example the implication is that the other person didn’t think that before, but has it as a new idea.