Is there any example of a metric space $X$ with more than two points such that the triangle inequality is always equality?
In any metric space with at least two points, the triangle inequality is an actual inequality. For if $x\neq y$ and thus $d(x,y) > 0$, then by the triangle inequality, $$ 0 = d(x,x) \leq d(x,y) + d(y,x) = 2d(x,y), $$ and by our hypothesis we must therefore conclude that $$ d(x,x) < d(x,y) + d(y,x). $$
Edit: removed some redundancies. The same conclusions hold for a nontrivial pseudometric.
Such an example does not exist ! Suppose that the metric space $X$ contains 2 points $x,y$ with $x \ne y.$
Then we have
$0=d(x,x) < d(x,y)+d(y,x)=2d(x,y)$
Perhaps no: $0=d(x,x)=d(x,y)+d(y,x)$; so $d(x,y)=0$ i.e. $x=y$.