" Turning on show suggestions in General Preferences doesn't help"
Wouldn't expect it to, but I'm guessing that was a typo. and you meant "off"
"(maybe because it's a month?)" is likely correct. Numbers will accept short form month names as part of a Date, and automatically format the entry as a date and time value. You can confirm your guess with a glance at the box at the lower left of the document window—if the auto-completed "March" cell shows a Date and Time string for midnight on March 1, 2019, your guess is correct.
Is the 'data format' for the cell(s) where this is happening set to "Automatic"? If so, try changing it to "text" for the whole column.
Regards,
Barry