No doubt they are only responsible for the airport to airport section of your travel legally speaking. You should ask them what is the position, it's always good to know for future flights.
That said from central Milan by taxi out to Linate was never the cheapest, despite the short distance, those Italian taxi-drivers loved their 'supplementos' which soon added up! I once had 5 extra charges on top of the fare (all legal) getting to Linate.
The train from Milan to Malpensa is pretty quick (30-ish minutes) and direct to the terminals, and not much different to the aforementioned short taxi ride :-)