Keep on is a phrasal verb taking a gerund clause as its complement; it means “continue [a course of action]”. It is an intensive version of plain keep in this sense.
Bob kept on interrupting me, so I couldn't finish. ... Bob continued to interrupt me
Laura kept on writing until the chapter was done. ... Laura continued to write
It often has the sense of “persevere”, as in the 60s-70s catchphrase
Keep on keeping on. ... Don't give up the struggle to achieve/survive
Keep ... as it is is not a fixed phrase but the transitive verb keep, meaning “maintain [something] [in a specified condition]” combined with the phrase as it is serving as the second complement with the meaning “in its present condition”.
Ryan wanted to change the second sentence, but we decided to keep it as it was.
Let's keep the schedule as it is for now. We can update it when we know more.
First of all, for similar phrases like this, I never like to say that they are interchangeable, because that implies that they connotate the same thing. There is usually a best choice to make. While the denotation of the phrases might be interchangeable, providing an incorrect connotation could lead to confusion on the part of the reader or listener of your sentence.
So let discuss connotation with an example.
In doing so:
Bob walked his dog. In doing so, he exercised.
The first phrase in doing so gives the impression that effect of his first action (walking the dog) wasn't at all intended to cause the second. He exercised, but this occurred only as a side benefit.
In this way:
Kathy finished watering her plants and set down the hose. In this way, the hose would be close by for next time.
In this example, the phrase in this way is used. It is implied that Kathy desired the results described in the second sentence. She wants the hose close by. However, it isn't a very strong desire. If her husband comes and stores the hose, she will just get it out again.
So that:
Fred went to Law school so that he could get a degree.
In this example, Fred performed the first action (law school) expressly for the purpose of achieving the second action (get a degree). This was his entire goal.
I'll change out the last example with a different phrase for clarity:
Fred went to Law school. In doing so, he could get a degree.
If you said this, a listener might be confused. They might ask "Did he want the degree?" or "Did he have another reason for going?" Maybe Fred's main purpose was to party with college students and blow off classes.
Conclusion:
Remember that changing out these phrases for each other will still make sense. It will still be grammatical. But it will connotate a different mindset on the part of the subject.
Best Answer
No, they mean different things.
"Keeping well" means "keeping healthy". So "I hope you're keeping well" is a slightly old fashioned greeting. It might be addressed to an older person (for whom being "well" might be in some doubt). I can't imagine a child using it with their friends.
To "keep doing well" is not an idiom. "Well" is the adverb related to "good". The speaker is hoping that the other person continues to do something in a good manner Eg.