One option is to create a new ID
column so that you have unique categories on the x-axis:
library(tidyverse)
df <- structure(list(Date = c("2016/01", "2016/02", "2016/03", "2016/04",
"2016/05", "2016/05", "2016/06", "2016/07"),
Value = c(2L, 3L,4L, 5L, 4L, 4L, 5L, 6L),
Label = c("A", "A", "A", "A", "A", "B", "B", "B")), .Names = c("Date", "Value", "Label"),
row.names = c(NA, -8L), class = c("tbl_df", "tbl", "data.frame"))
df %>%
unite(ID, Date, Label) %>%
ggplot(aes(ID, Value)) +
geom_col()

Alternatively you could use the Label
column to facet the data like so:
df %>%
ggplot(aes(Date, Value)) +
geom_col() +
facet_wrap(.~Label, scales = "free_x")
