Looks like 2 coincident truncated-hexagonal tilings (toxat), and indeed vertices, edges, and {12} all coincide by pairs.

Incidence matrix according to Dynkin symbol

β3x6x   (N → ∞)

both( . . . ) | 12N |  1  1  1 |  1 1 1
--------------+-----+----------+-------
both( . x . ) |   2 | 6N  *  * |  1 1 0
both( . . x ) |   2 |  * 6N  * |  0 1 1
sefa( β3x . ) |   2 |  *  * 6N |  1 0 1
--------------+-----+----------+-------
      β3x .   ♦   6 |  3  0  3 | 2N * *
both( . x6x ) |  12 |  6  6  0 |  * N *
sefa( β3x6x ) |  12 |  0  6  6 |  * * N

starting figure: x3x6x