Configuration analogues of projective spaces?

In a configuration, each point is incident to the same number of lines and each line is incident to the same number of points.

The Fano plane is a configuration, with 3 points on each line, and 3 lines on each point. It's possible to make the smallest 3D projective space with 15 points, 15 planes, and 35 lines.

Features:
1. Any three points not on a line define a plane with a configuration.
2. On any plane, the points and lines make a configuration.

My question -- are there 3D analogs of projective spaces involving configurations that are not projective spaces? For example is there a 3D set of points featuring lots of Desargues configurations?

-
I will have to dig for references, but I am pretty sure that uniqueness and coordinatization results are much stronger for dimension higher than $2$. –  zyx May 22 '13 at 22:01
I'd use “configuration” to denote any incidence configuration, and a more specific $(p_\gamma,\ell_\pi)$-configuration to denote those with the same number of incidences per point resp. line. I'm still unsure how you'd define “3D analogs”. You want something which is mostly but not completely a projective space, and the ways where it is and where it is not are left rather unspecified. Is that intentional? –  MvG May 23 '13 at 6:42
Yes, it's intentional. There are lots of configurations, but limited projective planes. I wondered if configurations could link up in some clever way. –  Ed Pegg May 24 '13 at 17:54