Anglica's explanation of the phrase 'on the up and up' is:
If you are on the up and up, you are making very good progress in life and doing well.
When I read this explanation, I wondered what was the difference between 'making very good progress' and 'doing well'. To an ESL, we would think the two would mean the same and we wouldn't dare to use them both at the same time.
Could I ask native English teachers to help me with this?
Often they could be used interchangeably, though 'doing well' can carry the idea of be financially successful, which the other doesn't IMO. Also, we often use phrases or words that have a similar meaning together, probably to cover all the bases.