Suppose I have a vector space $V$, and I identify $x\in V$ with $\lambda x\in V$, where $x\neq 0$ and $\lambda>0$, $\lambda\in\mathbb{R}$.
I'm confused about two things: (1) Can I define a norm on $V$, say $\|\cdot\|$ ?
Looking into this I see that $\|\lambda x\|=\|x\|$, but for a norm isn't it required that $\|\lambda x\|=|\lambda|\|x\|$ ? Or maybe because I've identified $x$ with $\lambda x$ then vectors of the form $\lambda x$ with $\lambda\neq 1$ are not permitted, so there is no problem ?
My second question (2). My "identification process" above reminds me of things like $\mathbb{Z}/n\mathbb{Z}$ where we have elements being equal modulo $n$ in that case. In my case we would have for example $(1,2)\equiv(2,4)$. Is there a quotient group like symbol for my space with identified elements?
A norm is a thing we sometimes define on a vector space. The quotient space you describe is not a vector space; it has no concept of addition. So one can't even begin talking about a norm on it.
The quotient space is naturally identified with unit sphere in $V$, since each equivalence class contains exactly one element with unit norm. This leads to a natural notion of metric on this space: just use the distance between the corresponding points of the unit sphere.