Yes, you can use the same match field to link to other tables so this would not be why it's not working for you.
You may want to use the controls below post A Answer to upload a screen shot of Manage | Database | relationships and then also describe exactly how the relationship is not working for you.
You might also want to use an actual date field set to the first day of a given month. Such a field will naturally sort in the correct chronological order so it can be a better option. Data formatting can be specified for such a field so that it displays only the month name and year so that it can look exactly like your current MonthID field.
And this calculation can take any date in "datefield" and compute the date for the first day of the month if you select "date" as the return type.
DateField - day ( DateField ) + 1