This is a really common situation with cars around the 2000-era. I have dealt with it on 164 and 166 and FIAT Coupe/Tipo (only Sedicivalvole)/Multipla. Just starting to see Stilos with it now, but other models (such as Punto Mk2) have escaped without it.
It is generally only the upmarket models that have the expensive-feeling soft-touch coating, so a Toyota Corolla won't have it (but Toyota Windom might). Toyota would probably still just laugh at you though, as I expect the Windom and other upmarket models are only sold in the Japanese domestic market (parts not widely available). On the other hand, VW/Audi sell quite a lot of replacement plastics and you can get replacement plastics for most models of FIAT/Alfa Romeo, too. How much would you like to pay?
The idea is that the coating gives a rubber-like feel, but as you point out, it destabilises over time (I don't know that sun really has all that much to do with it, since the earliest known failure was the 164 roof console 1988-93, and that never sees any sun at all).
You can use methylated spirits to remove the scratched, dirty, sticky part of the coating, which improves its appearance but tends to leave the surface still sticky. Good if you are trying to sell the car.
For a permanent solution, remove the coating completely by either prolonged rubbing with meths, or use an oven cleaner aerosol can and a scrubbing brush (the oven cleaner softens it and keeps it soft), or use wet-and-dry sandpaper and soapy water - or a combination of all three. Don't use paint stripper, as that etches the surface of the plastic.
I get a paint shop to mix up matt-finish lacquer, and then I paint the parts with several coats applied in a dry, dusty fashion so as to get a nice matt finish with an almost textured feel (but never the same rubber-feeling as the original). Matt black is easy, as there are spraycans readily available - I use Duplicolor.
There was a company called Foliatec in England who produced a spraycan for 'suede effect' interior paint. That would be the closest thing I have ever seen to the original, but restrictions on posting aerosol cans overseas made it impractical to obtain. And, it doesn't seem to be available any more (I wonder why!) Available in a wide selection of - two - colours, blue, or dark grey.
See the cached page at Foliatec Interior Colour Spray - Soft Touch (2040-2041) - Foliatec Car Styling Accessories - UK's
For a bit of general background on soft-touch coatings (and a specific discussion of materials), see The New Age of Waterborne Soft-Touch Coatings - Feature Articles - Paint and Coatings Industry
CASPortal - Softfeel
There is also a wonderful sales brochure at
Feel The Difference: Water-borne Soft Touch Coatings
Very upmarket, though of course you cannot actually buy the product - it just tells you about how important it is
It also talks about the problems of early soft-touch coatings when subjected to hot, moist air over a prolonged period and then to occasional contact with handcreams, insect repellant, etc. and of course, all these problems are solved now. Apparently.
Soft-touch coatings are not unique to car interiors, though my paint shop denies all knowledge. Mobile phone faceplates, laptop computers and mouse devices (you're not allowed to call them 'mice'...) made around the 2000-era came with a range of soft-touch 'rubber'-feel coatings (remember them?) If you ever manage to find any, you'll find they've gone sticky too - especially if kept in warm, dark conditions. I begin to wonder whether soft-touch coatings are a universal failure that the manufacturers are trying to forget.
I started a thread in the 166 forum a long time ago: http://www.alfaowner.com/Forum/alfa-...-plastics.html
Responses still come in dribs and drabs, and everyone seems to assume I still have the same problem even though I said I sorted it out!
Actually I solve this problem for other people quite often - I charge a few hundred dollars to strip and repaint all affected interior plastics. On something like a Multipla, it's a complex job because the whole dashboard has to come apart.