Making polynomials representing frequency of a character in a list

You can also use a combination of Merge and AssociationThread:

Merge[Total][AssociationThread[# -> x^Length @ #]& /@ l]

<|"a" -> x + x^2 + x^4 + 2 x^5,
"h" -> x^2,
"d" -> x^5,
"k" -> x^4 + x^5,
"r" -> x^5,
"v" -> x^5,
"b" -> x^4 + x^5,
"c" -> x^4 + x^5,
"s" -> x^5,
"u" -> x^5|>

I think @kglr's version is better, but here is my take

GroupBy[
 Flatten[
  Dot[
     Tally[#],
     DiagonalMatrix[{1, Power[x, Length[#]]}]
     ] & /@ l
  , 1]
 , First
 , Last@*Total
 ]