This is how I understand it:
My guess is that in the old days, the term "Regiment" in SIR
denoted a specific size of a unit.
So 1SIR really meant 1st Singapore Infantry Regiment, 2SIR meant
2nd Infantry regiment and so on. And in the old days we literally
only had a couple of regiments.
But these days, 1SIR means 1st Battalion of the Singapore Infantry
Regiment. 2SIR means 2nd Battalion of the same Singapore Infantry
Regiment.
So how big is a "Regiment" in Singapore?
My guess is that the "Regiment" no longer denotes a particular
size. It is merely some nostalgic hold on tradition to keep using
the term "SIR" for our units.
The term "Singapore Infantry Regiment" is more like the family
surname and doesn't denote a specific strength. So every battalion
in Singapore - irregardless of which Brigade or Division - is a
part of the BIG Singapore Infantry Regiment family.