When is $1\vec{u} \neq \vec{u}$? In a linear algebra textbook they define a vector space to be a nonempty set $V$ of objects that satisfy certain properties.
One of these properties is that $\forall\vec{u}\in V(1\vec{u}=\vec{u})$
The only way I can think of that property NOT holding would be if scalar multiplication were not defined for some object (not necessarily a vector).
Are there any other cases where multiplication of $n$ by 1 (the scalar multiplicative identity) would not equal $n$?
(not really sure what to tag this with)
 A: Think of it this way. If you did not explicitly make $1\vec{u}=\vec{u}$ into one of the vector space axioms, you would have no justifiable way to conclude that $1\vec{u}=\vec{u}$; this identity does not follow from the other vector space properties. (You can try to derive it, but you'll never get there using only the other axioms.)
Your intuition is good: you want $1\vec{u}=\vec{u}$ to be true in order to have an abstract vector space be in good correspondence to your perception of physical vector spaces and scaling. So you just need to put this relation $1\vec{u}=\vec{u}$  into the list of axioms.
To answer your question more directly, any mathematical structure that has $1\vec{u}\neq\vec{u}$ is, in my opinion, just being needlessly confusing, using the symbol "$1$" to mean something different from all common meanings of that symbol, or using the multiplication operation to mean something different from all common meanings. (Added later: as MarkS. points out in the comments, concatenation is indeed a common meaning for two items written adjacent to each other that I had overlooked.)
