How to aggregate categorical data in R?

As mentioned in the comments, table is standard for this, like

table(stack(DT))

         ind
values    Category.x Category.y
  Better           2          2
  Similar          1          2
  Worse            1          0

or

table(value = unlist(DT), cat = names(DT)[col(DT)])

         cat
value     Category.x Category.y
  Better           2          2
  Similar          1          2
  Worse            1          0

or

with(reshape(DT, direction = "long", varying = 1:2), 
  table(value = Category, cat = time)
)

         cat
value     x y
  Better  2 2
  Similar 1 2
  Worse   1 0

sapply(df1, function(x) sapply(unique(unlist(df1)), function(y) sum(y == x)))
#        Category.x Category.y
#Better           2          2
#Similar          1          2
#Worse            1          0

One dplyr and tidyr possibility could be:

df %>%
 gather(var, val) %>%
 count(var, val) %>%
 spread(var, n, fill = 0)

  val     Category.x Category.y
  <chr>        <dbl>      <dbl>
1 Better           2          2
2 Similar          1          2
3 Worse            1          0

It, first, transforms the data from wide to long format, with column "var" including the variable names and column "val" the corresponding values. Second, it counts per "var" and "val". Finally, it spreads the data into the desired format.

Or with dplyr and reshape2 you can do:

df %>%
 mutate(rowid = row_number()) %>%
 melt(., id.vars = "rowid") %>%
 count(variable, value) %>%
 dcast(value ~ variable, value.var = "n", fill = 0)

    value Category.x Category.y
1  Better          2          2
2 Similar          1          2
3   Worse          1          0