Why are every structures I study based on Real number?

why do I have to restrict the measurement standard to real number? Why not any ordered field?

You can use any ordered field, and the axioms will still make sense. The thing is, though, that most of our geometric intuition is built on the Archimedian property. But if we include this in our field, then it becomes a subfield of $\Bbb R$. In this case, we are really not restricting ourselves to $\Bbb R$ at all: it's the largest Archimdean ordered field!

Thinking about ordered fields that aren't Archimedian gets a little more weird. Are you prepared, for example, to have vectors $v,w$ such that $v$ and $w$ point in the same direction, and yet $vn$ is shorter than $w$ for every natural number $n$?

I mean, if we define $\|v\|=\langle v, v\rangle^{1/2}$ then indeed $(V, \|\, \|)$ becomes a normed vector space

No, not quite! Normally we want the norms to be in the base field, but the vector $(1,1)\in \Bbb Q^2$ would have a norm outside of $\Bbb Q$, with that definition. To fix things, you'd need something called a Pythagorean field, and that's enough to guarantee this definition of norm works.

If you don't ask for the norm to be in the base field, then you may not be able to carry out normalization, because dividing by the norm won't give you a vector in the field.

I know that the real number is the only (up to isomorphism) ordered field with the least upper bound property and the property that every increasing bounded sequence converges. But is that enough to justify that every metric spaces use real number? I want to be more convinced, then.

Yes, your intuition is along the right lines. The fact that $\Bbb R$ is "maximal" among Archimedian ordered fields makes it special. It's a smooth connected piece with no holes. In geometry this is important since it ensures that lines and circles cross where you expect them to. For example in $\Bbb Q^2$, you can find an example of a line and a circle that would intersect in $\Bbb R^2$, but they pass through each other in $\Bbb Q^2$ without touching.

Why not... anything else!!

Actually, geometers study generalizations of the norm in the form of bilinear forms over any field. The idea is that rather than focus on the norm, you instead focus on a (generalized) inner product. They can be quite different from what you're used to with run-of-the-mill normed real spaces. They can have, for example, nonzero vectors with length $0$ or even negative length.

Even more than that, "length" loses meaning totally when you're working in a field that isn't ordered: there's no such thing as positive or negative, there. Still, there is a huge theory for these types of spaces with (generalized) inner products.

You can define inner products for vector spaces and normed vector spaces over any ordered field.

You can define metrics over any ordered field: they are called generalised metrics.

Not any ordered field possesses the square root operation defined wherever $x ≥ 0$; rational numbers form an obvious counter-example known from ancient times. Yes, it is essentially the same problem as famous one faced by Pythagoreans, but in modern formulation: how to define number field to make things going well with geometry?

There can be numerous ways to tweak definition of an inner-product space, at expense of some nice properties (metric completeness, separability, symmetry of the inner product… ). We are even not obliged to compute the norm in the same field as the ground field of our vector space; the concept of valued field provides a possibility to do otherwise.

Real numbers are so popular not because they form “very correct” field, but because their arise from requirements deemed very natural in analysis: order, metric completeness, separability. Wherever you can sacrifice any of it, you are welcome to work over alternative fields.