ASIC means 'Application Specific IC', and that device is very application specific. If you want to build a 4 port Ethernet switch with it, it will do it, and only that. So yes, you need to use the 'ASIC' column for the MTBF figures from your reliability tables.
The fact that such figures are next to meaningless for each individual case is irrelevant for what you need to do. As a reliability bean-counter (bean-counter, perjorative term for accountant who does not need to understand what goes on underneath the figures), you only need to show that you have added up the correct numbers in the correct way.
The term 'ASIC' covers such a range of complexities, technologies and use-cases, the only realistic measure is whether the manufacturer can sell them successfully. If he can, it means they don't fall over too often. That means an asic intended for engine compartment use in a car is likely to be a lot more reliable than one intended for room temperature use in a cheap consumer item, and strangely enough, a lot more reliable than military use ones as well.