The statement “There is no x such that x+x = 1+1+1” is true in every extension of Presburger.* But that doesn’t mean there’s no extension where 1.5 is a number. I can make a model that satisfies Presburger in which 1.5 is a number. Namely, it’s the model just like the standard interpretation, but where we interpret the symbol “1” to refer to the number 1.5.
That model satisfies the Presburger axioms, and on that interpretation there is no x where x+x = 1+1+1 (remember to interpret “1” correctly!) and on that interpretation 1.5 is an element of the domain.
*It must be, since Presburger is complete, but my proof-fu is failing me when I actually try to construct the proof.