On one hand, according to wikipedia, it says that the pair $(X, \Sigma)$ is a measurable space, if $X$ is a set, and $\Sigma$ is a $\sigma$-algebra on $X$. To me, a "measurable space" means a space we can assign a meaningful measure to the measurable sets in the space.
On the other hand, it seems that the power set of any set is a $\sigma$-algebra on the set, so it means that if I take $X=\mathbb{R}$ and $\Sigma=2^{\mathbb{R}}$, then $(\mathbb{R}, 2^{\mathbb{R}})$ is a measurable space...but this is contrary to my current understanding (that for uncountable set $\mathbb{R}$, we usually need to find a smaller $\sigma$-algebra than $2^{\mathbb{R}}$, to define a meaningful measure).
What am I missing here?
You are not missing anything: the word measurable space might just be a little misleading. Measurable space only means: a set $X$ and a $\sigma$-algebra $\Sigma$ on $X$. Nothing more. Nothing less.