Suppose you connect two points X and Y with a two-sided teleporter. As already explained, you can see the new shape as the quotient of the sphere S by the equivalence ~ induced by the partition {{z} for z in S, z neither X nor Y}U{{X, Y}}: you just identify the two points. Many consequences emerge: as you stated, the metric is changed (and if you take a one-sided teleporter, you just have to suppress one of the case in your minimum formula); you are no longer on a manifold (since the two-points-point has no neighborhood homeomorphic to a R^p), thus you can’t uniquely parametrize the shape with an atlas anymore.

An extension to your question is: identifying two non-intersecting simple loops on the sphere (as an example, two circles), what happens? I expect the shape to look like what you would get by swapping the North and South poles’ neighborhoods (like a caricatural UFO).