I was just reading through the "What Is Geometry?" article (http://www.cut-the-knot.org/WhatIs/WhatIsGeometry.shtml). The final paragraph of that article says:

"There are many geometries. All of these share some basic elements and properties. Even finite geometries deal with points and lines and universally just a single line may pass through two given points. Thus I believe that a frequently used term "Taxicab Geometry" is a misnomer..."

If I understand correctly, this criterion is violated by spherical geometry, where some pairs of points are joined by an infinite number of lines (as with the meridians all passing through the poles on the earth's surface).

The comment about 'taxicab geometry' might be right, but maybe for a different reason.

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I vaguely remember reading somewhere that a geometry must satisfy:
dr^2 = dx^2 + dy^2 + dz^2 ...and so on for higher dimensions

That definition of a 'geometry' would include spherical geometry and exclude taxicab, but I don't know what other interesting 'geometries' it might exclude.

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By the way, that was an excellent article. I keep forgetting about this site and then stumbling on it again in web searches. Keep up the good work. :)