Firstly, the difference is not to do with formal/informal.
In "any of my X" I HAVE X.
In "any X of mine" I may or may not have X.
"any of my X" is common in everyday speech:
- If you need help please ask any of my staff.
- Were any of my ideas useful?
"any X of mine" is less common (as you discovered) and it gives greater emphasis to the word 'mine'. In fact when "any X of mine" is spoken the word "mine" is often stressed.
If any of my students would like to go to the concert would they please let me know.
If any students of mine would like to go to the concert would they please let me know.
- There is no difference in meaning at all between these two.
I wouldn't let any of my children go to that park.
- The meaning is clear. [If there were only two children then of course it would be "either of my children."]
I wouldn't let any child of mine go to that park.
- If I had any children (and I MAY have) I wouldn't let them go to that park.
Here there's an implied criticism of parents who DO let their children go to that park.
1) No-one has ever called any of my soldiers slovenly.
- The officer is not necessarily being proud here. He might even be amused.
2) No-one has ever called any soldier of mine slovenly.
- The officer is proudly taking the credit for having a smart platoon: smarter than yours perhaps.
NOTE! In sentence 1 it would be possible to stress the my and become as boastful as the officer in 2, though not as pompous. Perhaps it's this pomposity that you identified as formality. In certain situations "any X of mine" can sound rather superior.