Your understanding of the Planck length is incorrect. From the article you linked:
"According to the generalized uncertainty principle, the Planck length is in principle, within a factor of order unity, the shortest measurable length - and no improvements in measurement instruments could change that."
There is no claim that it is the smallest length possible, and there is certainly no claim that every length has to be an integral multiple of the Planck length.
What I stated above was an interpretation of the statement you cited. In essence, I (and many others) would posit that if you can't measure it, then you can never know it exists. So, the smallest length any object can have is the Planck length. And then I believe, that since you can't measure sub Planck lengths, everything you measure must be an integral number of Planck lengths. Obviously, one is free to disagree with this interpretation.
You can interpret it however you like, but your interpretation is not backed by logic or evidence. Does the uncertainty principle imply that position and momentum don't exist at the same time?
The uncertainty principle does not state that you can't measure both position and momentum; it states that you can't measure both accurately. So, as your accuracy of one increases, the accuracy of measurement of the other decreases.
On careful examination the physicist finds that in the sense in which he uses language no meaning at all can be attached to a physical concept which cannot ultimately be described in terms of some sort of measurement. A body has position only in so far as its position can be measured; if a position cannot in principle be measured, the concept of position applied to the body is meaningless, or in other words, a position of the body does not exist. Hence if both the position and velocity of electron cannot in principle be measured, the electron cannot have the same position and velocity; position and velocity as expressions of properties which an electron can simultaneously have are meaningless.
— Percy W. Bridgman
Reflections of a Physicist (1950), 90.
The strength and weakness of physicists is that we believe in what we can measure. And if we can't measure it, then we say it probably doesn't exist. And that closes us off to an enormous amount of phenomena that we may not be able to measure because they only happened once. For example, the Big Bang. ... That's one reason why they scoffed at higher dimensions for so many years. Now we realize that there's no alternative...
— Michio Kaku
Quoted in Nina L. Diamond, Voices of Truth (2000), 333-334.
> In essence, I (and many others) would posit that [p1] if you can't measure it, then you can never know it exists. So, [p2] the smallest length any object can have is the Planck length.
I think you missed the quote monjaro posted which I responded to "According to the generalized uncertainty principle, the Planck length is in principle, within a factor of order unity, the shortest measurable length - and no improvements in measurement instruments could change that."
i.e. Planck lengths is the shortest measurable length and thus you can't measure anything of smaller length, and so the smallest length any object can have is the Planck length.
"According to the generalized uncertainty principle, the Planck length is in principle, within a factor of order unity, the shortest measurable length - and no improvements in measurement instruments could change that."
There is no claim that it is the smallest length possible, and there is certainly no claim that every length has to be an integral multiple of the Planck length.