Under the FAA system there is is a clear distinction between a single engine certificate, and a multi engine certificate. The two are not interchangeable and nor can you just do an endorsement/type rating/differences training to add the other onto your current licence. The FAA system also differentiates between land & seaplanes for licence category.
This means that in the US & anywhere else that emulates the US licencing system there are 4 distinctive licences at each licence level for fixeds wing aircraft:
Private Pilot Single Engine Land (SEL)
Private Pilot Single Engine Sea (SES)
Private Pilot Multi Engine Land (MEL)
Private Pilot Multi Engine Sea (MES)
Commercial Pilot SEL
" " SES
" " MEL
" " MES
Air Transport Pilot SEL
" " SES
" " MEL
" " MES
At some licence levels there are credits granted for already holding a class of licence w.r.t. what must be tested for adding another class. You still have to do the licence test for that level.
Unlike JAA, Australia and similar jurisdictions in the US you could do a PPL test on MEL, a CPL on a MES and an ATP on a SEL, for example. Flying a MEL class you would be limited to PPL privileges, on a MES you would have CPL privileges and finally ATP privileges limited to SEL aircraft only.
In my case I hold FAA ATPs for SEL, SES & MEL but no FAA PPL or CPLs. None of my US licences allow me to fly MES.
Other jurisdications don't segregrate by category or class. Get some privilege at one eg PPL, then it transfers to any other licence you get subsequently. For example, pass PPL on a SEL, add a floatplane endorsment and then later pass a CPL test on a MEL you would finish with CPL privileges on SEL, SES, MEL & MES. The licence issue test is testing you for your competences to operate within the privileges of that licence level and not specifically for a type of aeroplane. So, in my case, my Oz ATPL has both single engine & multi engine types for land operations even though I never sat some ATPL level test in those. With the addition of my floatplane endorsement I get to fly single & multi engine seaplane under the umbrella of my ATPL.
JAR is more similar to the Oz system than the US. If I were to get a competency check in a single then I'd have that aircraft group added to my (currently multi engine only) UK ATPL.
Hmm....I wonder if the UK CAA would add floaty privileges based on my foreign licences as a paperwork + $ exercise?