Date values represented internally by numbers the common way are not suitable for Scatter-Charts if the month (or the year) should be shown on one axis. The simple reason is that months (and years) differ in length while the intervals on the axis need to be of same length (labels equidistant).
If the difference in length of the months definitely is not relevant, this is one of the rare cases where I might probably use the so-called Line-Chart. A Column-Chart may suit better:
Create an auxiliary column with an integer sequence contaíning the ordinals of the months, and a formula creating the wanted labels based on these integers as texts.
Not much more words: Study this attached example - and tell me if I misunderstood you.
(Use good formatting like YYYY-MM instead of outdated MM/YYYY. Reserve the slash for fractions.)
===Edit 2020-03-16 about 12:50 UTC===
The attachment announced in my recent comment: ask233774monthsAsAxisLabels_1.ods
===Edit 2020-03-16 about 19:45 UTC===
Sorry! Just came back to this thread and noticed that the second attachment I had made wasn’t the one I had intended.
Now (hopefully) the correct attachment: ask233774plotWithMonthNames_1.ods