Subsetting under multiple conditions

You need to group by Transmitter (missing from your attempt) and ensure that both values are in each group of Season.

dplyr

library(dplyr)
out <- dat %>%
  group_by(Transmitter) %>%
  filter(all(c("Winter1", "Winter2") %in% Season)) %>%
  ungroup()
out
# # A tibble: 12 x 5
#    Transmitter    Batch.location Location.Dispersion Season   Freq
#    <chr>          <chr>          <chr>               <chr>   <int>
#  1 A69-1602-59814 Lemmer         Lemmer              Winter1  1961
#  2 A69-1602-59814 Lemmer         Lemmer              Winter2  2075
#  3 A69-1602-59815 Lemmer         Lemmer              Winter1   310
#  4 A69-1602-59815 Lemmer         Lemmer              Winter2     1
#  5 A69-1602-59826 Lemmer         Lemmer              Winter1   997
#  6 A69-1602-59826 Lemmer         Lemmer              Winter2  3475
#  7 A69-1602-59828 Lemmer         Lemmer              Winter1   988
#  8 A69-1602-59828 Lemmer         Lemmer              Winter2  2991
#  9 A69-1602-59831 Lemmer         Lemmer              Winter1  3147
# 10 A69-1602-59831 Lemmer         Lemmer              Winter2  6155
# 11 A69-1602-59835 Lemmer         Lemmer              Winter1    76
# 12 A69-1602-59835 Lemmer         Lemmer              Winter2  1921

And from here you can use n_distinct or something else to count the unique Transmitter values you need.

summarize(out, n = n_distinct(Transmitter))
# # A tibble: 1 x 1
#       n
#   <int>
# 1     6

or just

length(unique(out$Transmitter))
# [1] 6

base R, option 1

ind <- ave(dat$Season, dat$Transmitter,
           FUN = function(z) all(c("Winter1", "Winter2") %in% z)) == "TRUE"
ind
#  [1]  TRUE  TRUE  TRUE  TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE  TRUE  TRUE FALSE  TRUE  TRUE FALSE  TRUE  TRUE FALSE
# [21] FALSE FALSE  TRUE  TRUE FALSE
dat[ind,]
# ...

length(unique(dat[ind, "Transmitter"]))
# [1] 6

The == "TRUE" use of a character "TRUE" is because ave forces the return value to be the same class as its first argument, which is dat$Season. Internally it calculates logical but is coerced to string afterwards. (Just run the ave(..) without ==... to see this in action.)

base R, option 2

sum(aggregate(Season ~ Transmitter, data = dat,
              FUN = function(z) all(c("Winter1", "Winter2") %in% z))$Season)
# [1] 6

split by season, then using intersect and length.

with(dat, 
     do.call(\(...) intersect(...), unname(as.list(split(Transmitter, Season))))
     ) |> length()
# [1] 6

Or using table and count rows with rowSums equal to two.

with(dat, table(Transmitter, Season)) |>
  (\(x) x[rowSums(x) == length(unique(dat$Season)), ])() |>
  nrow()
# [1] 6

(Dispersion$Season == "Winter1") & (Dispersion$Season == "Winter2") is looking for rows where Season is both "Winter1" and "Winter2" in the same row (at the same time), which is why that doesn't work. Since you're using dplyr, I would do it this way:

Dispersion %>%
  group_by(Transmitter) %>%
  filter(all(c("Winter1", "Winter2") %in% Season)) %>%
  ungroup() %>%
  summarize(n_trans = n_distinct(Transmitter))
# # A tibble: 1 × 1
#   n_trans
#     <int>
# 1       6

Another base solution:

sum(by(dat$Season, dat$Transmitter, FUN = \(x) { all(unique(dat$Season) %in% x) }))

# [1] 6