How to plot month numbers in gnuplot 5.4?

I'm not sure whether I fully got your intention. I understand you want column 1 as xtic labels. However, there is a difference between just taking the column 1 as xtic labels or interpreting column 1 as date/time and scaling the x-axis accordingly and gnuplot would take care automatically about the xtic labels. For the first case, the width of the graph might not be wide enough to show all labels without overlap, so, I reformatted date. Check help strptime and help strftime. By the way, help tm_mon says it needs input in seconds, not in your date format as it is in $1, there for use strptime().

I understand you want the yrange ranging from 1 to 12 for the months of a year. But where is the data you want to plot? Maybe the following examples is a starting point to better find out what you really want.

Code:

### plotting dates...
reset session

$Data <<EOD
2019/01/01
2019/02/01
2019/03/01
2019/04/01
2019/05/01
2019/06/01
2019/07/01
2019/08/01
2019/09/01
2019/10/01
2019/11/01
2019/12/01
2020/01/01
2020/02/01
2020/03/01
2020/04/01
2020/05/01
2020/06/01
2020/07/01
2020/08/01
2020/09/01
2020/10/01
2020/11/01
2020/12/01
EOD

myTimeInputFmt = "%Y/%m/%d"
myTimeOutputXFmt = "%Y\n%m\n%d"
set xlabel "Date"
set format x myTimeOutputXFmt   # to get enough space for the 3 lines of the label

set ylabel "Month"
set yrange [0.5:12.5]
set ytics 1

plot $Data u 0:(tm_mon(strptime(myTimeInputFmt,strcol(1)))+1): \
             xtic(strftime(myTimeOutputXFmt,strptime(myTimeInputFmt,strcol(1)))) \
             with points pt 7 lc "red" title 'data'
### end of code

Result: