I have found 2 issues with the xml which I hope can help to narrow down the source of the bug with repeatedly asking to select a new profession when the player gets to level 5 or level 10.
This was identified and tested with both 12.9 inch iPad Pro and an iPhone 13 pro running version 1.5.6.26.
When my character reached level 10 in mining I selected Excavator. And the next night I was asked again, like others have reported.
I looked at the xml and found two interesting things.
1. in the <professions>...</professions> section instead of adding the new profession a single time int added the the new profession twice. It differed from the previous nights save by having <int>22</int><int>22</int> I presume the code should only have added <int>22</int> a single time.
2. The proximate cause for repeating the messages is the xml section <newLevels>...</newLevels>
in the files which causes the question to be repeated the prettified xml reads:
<newLevels>
<Point>
<X>3</X>
<Y>10</Y>
</Point>
</newLevels>
In the previous night's file, before maxing the skill, the corresponding entry is a piece of malformed xml containing only the end tag: </newLevel>. My xml linter rejected the file but the game apparently accepts bare closing tags without complaint since that file works normally.
I patched the problem in my own current file by replacing the section containing the skill+level tuple with an empty section <newLevels></newLevels>, and by removing the duplicated profession selection.
I hope this helps.
For users who are brave enough to fix the problem by editing their own save files. I say, keep a safe backup before touching the file. And yes, you can get it to stop asking by removing the <Point>...</Point> information in the <NewLevels> section. You can try doing that alone, or also, as I did, find a duplicate <Int>..</Int> in the <professions> list and cleaning that up.