I had this very thing the only last week.
It's not the battery as it's only a year old and the Stop/Start works perfectly.
I connected my PC with
MultiEcuScan and found some error codes and cleared them. Then, I tried again, and more error codes came up plus the warnings again ........ as you describe.
The Fiat 500 is a computer with an engine and wheels. When a computer throws its toys out of the pram, you switch it off and reboot. Rarely is it a hardware problem.
What I did, was to disconnect the battery and go and have a cup of tea.
Half an hour later, I reconnected the battery (and reset the date and time) and all the errors had gone and none of the warnings came up again.
The car has worked perfectly ever since.
Therefore, FIRST thing to do is to disconnect the battery.
Second, have a cuppa.
Third, reconnect the battery, and fourth is reset the date and time.
Fifth is to check if all is well.
If not, repeat the above.
Good luck,
Mick.