Tingle!
Note the measuring circuit ... it looks kind of familiar. But where's the aluminium foil?
The earth connection is for EMC purposes and has no effect on the isolation between the input (mains) and the output.
Yes, you're right ... for Class II. If you can get hold of a decent circuit diagram for a SMPS, you'll often find the "earth" (protective earth conductor of the mains cable) connected in-between filter capacitors across the mains input (that is, early on in the circuit - and in just the one place). So a Class II situation, but (confusingly, perhaps) with a three-prong mains input.
But occasionally you may instead see it connected straight across to the "zero volts" output, with other components on the output side also connected to it. A Class I situation, then.
If the "brick" only has two mains input connections (as shown in the link above, for example), then it has to be Class II (of course). Otherwise, one way of distinguishing between a Class I and Class II "brick" is by looking at the diagram (which you probably won't have). Another way is to see what your multimeter says (ohms) between the zero volts output connector and the earth prong on the input side.
If you're lucky (and as Mike says) you might also find a Class II "double box" symbol on the labelling ... but personally I wouldn't (don't) pay too much attention to that (or indeed its absence), preferring instead to trust my old multimeter.
