Why can only like terms be combined through addition and subtraction?

This is quite clear when you add measurements, for example, lengths. If you just go ahead and add the numbers, without considering differences in units, you get DIFFERENT RESULTS, depending on the units used. For example, if you add 1 m + 2 m, you get 3 m - but if you express this as 1000 mm + 2 m, you get 1002. 1002 what? Anyway, this isn't the same answer as before, whether you decide that the 1002 represents meters, or mm. So, something is obviously very wrong; there is no way to make sense out of the calculations - unless you FIRST convert to the same units.If you try to add different units ("unlike terms"), for example meters and square meters, the situation gets even worse.