Yes and no.
There are differences, but these S1, 2 and 3 numbers don't always relate directly to one of oil industy standards, they are Fiat standards.
The oil industry use various standards and oil is made to to those oil industry standards.
ACEA (European Automobile Manufacturers' Association)
SAE (Society of Automotive Engineers)
API (American Petroleum Institute)
JASO (Japanese Automotive Standards Organization)
This oil is then produced in various viscosities. (common to all standards)
0W30, 5W40, 10W40 etc
Then oil in quality. (ACEA qaulity standards below, other standards use a different code)
A – oil designed for petrol engines.
B – oil designed for diesel engines in private cars.
C – for light engines equipped with catalytic converters or particulate filters.
E – for commercial vehicles and trucks.
Then different performanes. (again ACEA standards)
Three category combinations for petrol and diesel engines: A3/B3, A3/B4, A5/B5.
Five categories for vehicles with a pollution control device: C1, C2, C3, C4, C5.
Four categories for commercial vehicles and trucks E4, E6, E7, E9.
Now Fiat might say the engine oil for a certain Fiat engine needs to be 5W40 ACEA C3.
There are probably dozens of 5W40 ACEA C3 oils on the market but not all might be suitable or have been tested, so Fiat set their standard so oil companies can make it to that and helps owners identify the right one.
So they add that it must meet one of their specs, eg 9.55535-S2.
So this 9.55535-S2 number helps to select the correct 5W40 ACEA C3 oil that Fiat wants you to use in that particular engine.