It drives as normal and the speedo works, could be a speed sensor but would the ABS light not be off until you start to move, I haven't done a lot of work on Fiats , I haven't listened for the ABS pump sound, the snow has been heavy here so I have not managed to do much looking at anything, I have two code readers plugged them in, but none show further than the engine, no ABS or any other info.
The system will self check all the circuits when the ignition is switched on.
If if finds an open one, it'll put the light on right away without having to drive it.
If the circuits are ok, but there's a iffy signal being read, then you would expect the light to come on when it detects that iffy signal, like when it's moving.
Normal, generic codes readers will only read the engines ECU and won't connect with chassis or body control modules. You need something that can read the chassis modules that the ABS is part of.
Common ABS issues tend to be wheel speed sensors failing, bad circuits in the ABS pump/module and weak or failed pump motors.
Wheels speed senors aren't that expensive or hard to fit and pump/modules with their motors can be repaired and rebuilt fairly cheaply by a specialist like BBA Reman.
You can sometimes get a fault from a related input sensor that throws the ABS into a tizzy, like speed sensor input or brake related circuit like the fluid level or brake light switch.
I chased a real headache of a fault with the ABS in my Alfa Spider, turned out the SRS seat harness with iffy (and the Mrs is shorter than me) every time the drivers seat was moved it caused an open circuit and buggered up some input to the ABS for some reason, why I never found out, though suspect the ABS system was linked to the seat belt pre tensioners somehow.
Save yourself a lot of guess work and get a proper diagnostic check of the ABS module done.