The computer will sense alternator charging, not by battery voltage, but by a separate signal wire. This is probably the small wire on the back of the alternator. From Haynes, on the 500 this feeds to the engine management computer, which will tell the body computer, but on the Panda, it feeds to the body computer. So there are choices. I don't have a wiring diagram for the Stilo.
Before computerisation, this wire was fed battery voltage, and earthed through the alternator when not charging, the earth path being broken when the alternator gave an output. I would expect this to still be the same, rather than having an output from the alternator through that wire.
This would need to be tested though. By disconnecting this, and with ignition on, but engine not running, see if there is a voltage at that wire end. A voltage there shows it is workiing as older systems. No voltage may mean no output from the computer, or it is expecting an input to the computer. Testing that is more difficult, and probably involves a voltmeter on the terminal when the wire is connected to the alternator, and with either engine running, or alternator spun by hand, if the drive belt is off. Access is difficult, and potentially risky, with the possibility of shorting things, with burns and fire resulting, or worse, getting caught in moving parts. So take extreme care.
If the alternator is just not sending the correct signal, its internal diode pack is failing. A replacement alternator is the way to go for this.
On both 500 and Panda the main output from the alternator seems to connect to the starter, just as a connection point, to feed back directly to the battery.
You say you know it is charging, presumably with 14v or so at the battery, and the battery not going flat. On the smaller vehicles, Panda, 500 and Punto, the main earth cables corrode internally, creating problems, usually showing as a steering fault initially. No reasson to suppose Fiat used better cables on a Stilo, so would be a good idea to carefully check the earth cable, from battery, to body and engine. Not just connections, but for continuity and resistance. Starter and steering require big currents, whereas charging does not, so a charge can be 'felt' despite poor connections.
The steering will try to work as long as it has enough power. It does not take a message from the body computer about whether to be lazy, it just uses what is available via its own fat supply cable.
The warning lamp is controlled presumably from the body computer, which is not sensing sufficient alternator output. It is doubtful that there are multiple faults, so I think the main cables and their connections are the places to look.