Is my kids' alphabet mat properly grouped by colors?

My kids have an alphabet mat to play with, something like this:

Image may be NSFW.
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After months with the tiles of the mat randomly placed, I got tired and placed all the tiles of the mat grouped by sections according to their background colors. So, if the letters represent the background color, I got a mat like this:

AABBCDDDEABBCCCDEEABCCCCDDEAACCCDDEEAAAACCCCEAAAAAACCC

So, for colors A, B, C, D and E there is always a way to connect all the tiles with the same background color either horizontally or vertically in the mat. That's what I call a mat properly grouped by colors. You can see the groups for the previous example in the following tables:

AAAAAAAAAAAAAAAA  BB BB B    C   CCC  CCCC  CCC    CCCC      CCC     DDD      D      DD     DD        E       EE        E       EE        E

Also, there is only one group for every color, so this would not be valid:

ABAABA

Because color A tiles are not grouped in only one group. This also would not be valid because the tiles do not connect either horizontally or vertically:

ABBA

The challenge

Given a 2-dimensional array of characters in the printable ASCII range (does not need to be a square one as long as the size of both dimensions are equal to or greater than 1), check if the array represents a mat properly grouped by colors (each different character in the array represents a different color). Input may be in any reasonable format as long as it represents a 2-dimensional array of characters (2D char array, array of strings of the same length, and so on), and output must be a pair of truthy and falsey values (0/1, 't'/'f', true/false, whatever as long as something is returned and the return values are consistent across inputs).

This is code-golf, so may the shortest program/function/method/lambda for each language win!

Examples

A    truthyABAB   truthyABBA   falseyABCDE    truthyABCDC    falsey**::dd22***:d222*:::::22    truthy$$$%%%&&$$%%&&&&&&$$$%&&    falseyAABBCDDDEABBCCCDEEABCCCCDDEAACCCDDEEAAAACCCCEAAAAAACCC   truthyAABBABBAAAAA    truthyAAABAAAAAAAA    truthy

My mat properly grouped by colors

Image may be NSFW.
Clik here to view.

(I still have to fix those borders...)