This solution is admittedly a bit of a hack, but I think it gets around the issue of all bins having the same value. Distinct counts of the first variable are checked. If all counts are the same (length(n) = 1), a dummy data frame is created with the first value of x repeated n + 1 times and bound to the data frame to plot. Then, in the plot, the newly added data is "hidden" by setting x and y limits in coord_cartesian. I'm not sure how well this generalizes, but I hope it provides a work-around or inspires a different way to solve your issue.

library(tidyverse)
df = data.frame(x = c("a", "b"),
y = c("A", "B"))
# determine distinct count(s)
n = df %>% count(x) %>% distinct(n) %>% pull(n)
# establish new data frame to plot
d = df
# if all values have the same count (length = 1)
# create a dummy data frame with counts n + 1
# bind to the data frame to plot
if(length(n) == 1) {
dummy = data.frame(x = rep(df$x[1], n + 1),
y = rep(NA, n + 1))
d = bind_rows(df, dummy)
}
# resulting data frame to plot
d
#> x y
#> 1 a A
#> 2 b B
#> 3 a <NA>
#> 4 a <NA>
# create plot and "hide" the dummy data via coord_cartesian
ggplot(d) +
geom_tile(aes(x = x, y = y, fill = after_stat(count)), stat = "bin2d") +
coord_cartesian(xlim = c(0.5, n_distinct(df$x) + 0.5),
ylim = c(0.5, n_distinct(df$y) + 0.5),
expand = F) +
scale_fill_binned()