According to your proposed definition, the sequence $1,2,3,\dots$ would be Cauchy in $\mathbb{R}$, witnessed by the sequence of open sets $U_n = (n,\infty)$.
Edit: Let me incorporate some information from the comments to make this a more complete answer.
As another example of why your definition is unsatisfactory: In $\mathbb{R}^2$, any sequence $(a_n,0)$ of points on the $x$-axis is Cauchy, witnessed by the sequence of open sets $\mathbb{R}\times (-1/n,1/n)$.
The fact that completeness and Cauchyness are not topological properties can be formalized by the fact that there are generally many different metrics compatible with a given topology, and these different metrics can induce different notions of completeness and Cauchyness. Looked at a different way, homeomorphisms preserve all topological properties (I would take this to be the definition of a topological property), but homeomorphisms between metric spaces do not necessarily preserve completeness (see the examples in btilly's and Andreas Blass's comments).
On the other hand, the notion of completeness actually lives somewhere in between the world of topological properties and metric properties, in the sense that many different metrics can agree about which sequences are Cauchy. It turns out that they agree when they induce the same uniform structure on the space. And indeed, completeness can be defined purely in terms of the induced uniform structure, so it's really a uniform property.
There is one class of spaces in which topological property and uniform properties coincide: a compact space admits a unique uniformity. So you could call completeness a topological property for compact spaces. But in a rather trivial way, since every compact uniform space is automatically complete.
It could be worthwhile to view compactness as the proper topological version of completeness, i.e. the topological property that comes the closest to agreeing with the uniform/metric property of completeness.
The metric function on a metric space intuitively measures the shortest distance between any two points of the space. The triangle inequality has a very simple intuitive meaning: This shortest distance between two points shouldn't shrink if we go via a third point. That would violate how "shortest distance" ought to work.
Euclidean spaces (with some standard metric, like Pythagoras) are the archetypal examples of metric spaces, and one could argue that the notion of metric space is intended to be a generalisation of the notion of Euclidean space (I don't know the actual history here, but it seems reasonable). But it is just that: a generalisation. There are many things that metric spaces can do that Euclidean spaces can't. This is what makes it a useful generalisation: we lose some specificity, so we can't prove all the same things. But on the other hand, the results we do find apply to many non-Euclidean contexts.
Ultimately, that's the reason for the triangle inequality: it works to make a useful generalisation. We could forgo it, and have much more general spaces. This way our results would apply much more broadly. On the other hand, we wouldn't get as many proven results. This is always a trade-off. And the particular trade-off we have made the way we have defined metric spaces has been popular and fruitful.
Best Answer
A $2$-torus is a topological space $X$ that is homeomorphic (one-one continuously equivalent) to the surface $S$ of a doughnut in three-space. For all practical purposes we may require that $X$ is diffeomorphic (one-one differentiably equivalent) to $S$.
A Riemannian metric on $X$ is a law that allows to measure the length of (continuously differentiable) curves on $X$. This law is expressed "in local coordinates" $(u_1,u_2)$ by means of a quadratic expression $ds^2=\sum_{i,k} g_{ik} du_i du_k$. For a curve $\gamma:\ t\mapsto u(t)$ $\ (a\leq t\leq b)$ its length is then given by $L(\gamma)=\int_a^b\sqrt{\sum_{i,k} g_{ik} \dot u_i \dot u_k}\ dt$. When $X=S$ with the metric "inherited from" ${\mathbb R}^3$ then the formulae from treble's answer apply.
There is a very deep theorem about Rienmannian $2$-tori in general. It says the following: If $X$ is a Riemannian $2$-torus then there is a lattice $\Lambda$ in ${\mathbb R}^2$ (with fundamental parallelogram $P$) and a function $\rho:{\mathbb R}^2\to{\mathbb R}_{>0}$, periodic with respect to $\Lambda$, such that $X$ can be regarded as ${\mathbb R}^2/\Lambda$ ("$P$ with parallel edges identified"), provided with the Riemannian metric $ds=\rho(z)|dz|$, where$|dz|:=\sqrt{|dx^2+dy^2}$.
If the function $g$ is actually a constant then we say that the metric on $X$ is "locally euclidean". But note that the global metric structure of $X$ depends also on the shape of $\Lambda$, so there is an infinity of different "locally euclidean" $2$-tori.
In the case of a "real" doughnut embedded in $3$-space the corresponding lattice $\Lambda$ is orthogonal, and the function $g$ is $\Lambda$-periodic, but not constant.