What you should be aware of is that strictly speaking $\mathbb R$ is not a metric space. $\mathbb R$ is just a set (of real numbers), in order to be a metric space one need to have a metric too.
Normally one assumes that we have the metric $d(x,y) = |x-y|$, which makes $(\mathbb R, d)$ a metric space. Normally one sloppily just writes $\mathbb R$.
You could define another metric, for example the discrete metric $e(x,y)$ (that is $0$ if $x=y$ and $1$ if $x\ne y$). This makes $(\mathbb R, e)$ a metric space too.
There's no requirement of a metric to have the homogenity property. In fact there's no requirement for a metric space to be linear to start with, that is scaling the element does not need to make sense. All the other properties of a metric is met by the discrete metric so there's actually nothing that hinders it (ie $(\mathbb R, e)$) from being a metric space.
Note that it's the same with normed spaces. They aren't just a set, it's a set together with a norm - so $\mathbb R$ is strictly speaking not a normed space, but $(\mathbb R, |\cdot|)$ is. What we can say about a norm on $\mathbb R$ need to be of a specific form, for we have $||x|| = ||x\cdot x|| = |x|\cdot||1||$, that is every norm is a scaled absolute value.
Note also that to be really strict we shouldn't call $\mathbb R$ a linear space. We should then include the operations of scaling and vector addition. However this often becomes impractical - instead the convention is that unless explicitly stated one assumes certain operations to be used (we assume that addition is the normal addition, multiplication is the normal multiplication, scaling too, absolute value is the norm and so on).