If the radio is not turning on it means it's not getting the voltage. On every aftermarket radio I've seen there was a diagram showing the proper connection.
The fact that the connectors from the loom physically match the ones on the radio does not mean the cables are in the correct positions. Before the test check that the fuse in the radio is OK, you could have blown it earlier...
You have two possibilities: play with the cables or buy a suitable FIAT - ISO adapter on ebay or equivalent. Or a local shop. While buying, check that it will fit your 20 years old Stilo. Things could have changed in the last 20 years, newer FIATs can have other looms / connectors. I was buying one for my 2011/2012 Ducato a couple of years ago, it looked like this one:
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