R data cleaning in seqence

If you want to identify where the mistakes occur, here might be option using ave + cummax + cummin with base R

within(
  df,
  err <- ave(
    Mileage,
    CarID,
    FUN = function(x) replace(cummax(x) == rev(cummax(rev(x))), length(x), 0) + replace(cummin(x) == rev(cummin(rev(x))), 1, 0)
  )
)

which gives

    CarID FuelTransactionDate Mileage err
1  AAA555          05.01.2019    5060   0
2  AAA555          30.01.2019    7800   0
3  AAA555          14.02.2019    9100   0
4  AAA555          24.02.2019    9900   0
5  AAA555          07.04.2019  101110   1
6  AAA555          12.04.2019   12500   0
7  AAA555          15.05.2019   13000   0
8  AAA555          09.06.2019   13422   0
9  BBB788          15.05.2018   15000   0
10 BBB788          04.06.2018   15200   0
11 BBB788          19.06.2018   16150   0
12 BBB788          16.07.2018     100   1
13 BBB788          27.08.2018   17500   0
14 BBB788          10.09.2018   17999   0
15 BBB788          13.10.2018   18200   0
16 BBB788          02.11.2018   18555   0

Here's a method showing how to identify outliers and then fill them in using approx. I start by looking for decreases in mileage - you can put whatever additional conditions you want to check in the if_else to identify outliers:

dd %>%
  group_by(CarID) %>%
  dplyr::mutate(
    # replace mistakes with NA
    MileageNA = if_else(Mileage < lag(Mileage, 1, default = 0), NA_integer_, Mileage),
    # fill in missing values with approx
    # approx is nicely robust in case you have multiple mistakes in a row
    #   See the help page and the rule argument to control behavior
    #   in case you have mistakes as the first or last observations
    MileageCorrected = approx(MileageNA, xout = 1:n())$y
  )
# # A tibble: 16 x 5
# # Groups:   CarID [2]
#    CarID  FuelTransactionDate Mileage MileageNA MileageCorrected
#    <chr>  <chr>                 <int>     <int>            <dbl>
#  1 AAA555 05.01.2019             5060      5060             5060
#  2 AAA555 30.01.2019             7800      7800             7800
#  3 AAA555 14.02.2019             9100      9100             9100
#  4 AAA555 24.02.2019             9900      9900             9900
#  5 AAA555 07.04.2019           101110    101110           101110
#  6 AAA555 12.04.2019            12500        NA            57055
#  7 AAA555 15.05.2019            13000     13000            13000
#  8 AAA555 09.06.2019            13422     13422            13422
#  9 BBB788 15.05.2018            15000     15000            15000
# 10 BBB788 04.06.2018            15200     15200            15200
# 11 BBB788 19.06.2018            16150     16150            16150
# 12 BBB788 16.07.2018              100        NA            16825
# 13 BBB788 27.08.2018            17500     17500            17500
# 14 BBB788 10.09.2018            17999     17999            17999
# 15 BBB788 13.10.2018            18200     18200            18200
# 16 BBB788 02.11.2018            18555     18555            18555

I just put my comments in an answer just to better show the outputs: Here is the code:

library(dplyr)
library(ggplot2)

df %>% group_by(CarID) %>% 
  summarise(min = min(Mileage),
            max = max(Mileage))

df %>% group_by(CarID) %>% mutate(rate = Mileage/lag(Mileage, n = 1, default = NA)) # if < 1 then the previous value was higher.
df %>% group_by(CarID) %>% mutate(rate = Mileage - lag(Mileage, n = 1, default = NA)) # if < 0 then the previous value was higher.

ggplot(data = df, aes(x = CarID, y = Mileage)) +
  geom_boxplot()

Some outputs you can work with:

Using dplyr to remove case when n < n+1 CAUTION you might want to remove outliers before!

    > df %>% 
+   group_by(CarID) %>% 
+   mutate(rate = Mileage - lag(Mileage, n = 1, default = NA)) %>% 
+   filter(rate > 0)
# A tibble: 12 x 4
# Groups:   CarID [2]
   CarID  FuelTransactionDate Mileage  rate
   <chr>  <chr>                 <int> <int>
 1 AAA555 30.01.2019             7800  2740
 2 AAA555 14.02.2019             9100  1300
 3 AAA555 24.02.2019             9900   800
 4 AAA555 07.04.2019           101110 91210
 5 AAA555 15.05.2019            13000   500
 6 AAA555 09.06.2019            13422   422
 7 BBB788 04.06.2018            15200   200
 8 BBB788 19.06.2018            16150   950
 9 BBB788 27.08.2018            17500 17400
10 BBB788 10.09.2018            17999   499
11 BBB788 13.10.2018            18200   201
12 BBB788 02.11.2018            18555   355

The data:

df <- structure(list(CarID = c("AAA555", "AAA555", "AAA555", "AAA555", 
                               "AAA555", "AAA555", "AAA555", "AAA555", "BBB788", "BBB788", "BBB788", 
                               "BBB788", "BBB788", "BBB788", "BBB788", "BBB788"), FuelTransactionDate = c("05.01.2019", 
                                                                                                          "30.01.2019", "14.02.2019", "24.02.2019", "07.04.2019", "12.04.2019", 
                                                                                                          "15.05.2019", "09.06.2019", "15.05.2018", "04.06.2018", "19.06.2018", 
                                                                                                          "16.07.2018", "27.08.2018", "10.09.2018", "13.10.2018", "02.11.2018"
                               ), Mileage = c(5060L, 7800L, 9100L, 9900L, 101110L, 12500L, 13000L, 
                                              13422L, 15000L, 15200L, 16150L, 100L, 17500L, 17999L, 18200L, 
                                              18555L)), class = "data.frame", row.names = c(NA, -16L))

Maybe you can try removing outliers?

Q <- quantile(dataframe$Mileage, probs=c(.25, .75), na.rm = FALSE)
eliminated<- subset(dataframe, dataframe$Mileage > (Q[1] - 1.5*iqr) & dataframe$Mileage < (Q[2]+1.5*iqr))