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This Grünbaumian tesselation indeed has all elements in coincident pairs. Consider a hollow triangle: then we need for alternating haxagons circling around at least a double wrap.

Incidence matrix according to Dynkin symbol

x6o6/5x∞*a   (N,M → ∞)

. .   .    | 6NM |   2   2 |  1  2  1
-----------+-----+---------+---------
x .   .    |   2 | 6NM   * |  1  1  0
. .   x    |   2 |   * 6NM |  0  1  1
-----------+-----+---------+---------
x6o   .    |   6 |   6   0 | NM  *  *
x .   x∞*a ♦  2M |   M   M |  * 6N  *
. o6/5x    |   6 |   0   6 |  *  * NM