I think this is a feature.
The definition of the MONTH() and YEAR() functions require their single argument to be a date value - that is, a numerical value which can be formatted as a date. I think in giving them a text argument, you are relying on the same interpretation rules being followed to convert your text into a date as happens when you type a text string as a date into a cell. And those rules changed in version 3.6.2: some formats that worked previously no longer do so. This sort of thing is very locale-dependent, so I won't try to be definite about what is happening in your case. Your first question should be what happens if you enter 10/2013 into an unformatted cell: do you see a date value, perhaps 1 October 2013, or do you see the text string you entered?
Note that the date acceptance patterns can now be adjusted at Tools | Options... | Language Settings | Languages | Language of | Date acceptance patterns. Note also that the ISO format YYYY-MM-DD works in all locales, so is most reliable.
It occurs to me that you would never want to use a literal text value as in your example: 10 would be a lot easier to type than =MONTH("10/2013")! So you must be wanting to refer to the contents of a cell or some other expression, of course. In that case, you would presumably be best advised ether to store proper (numerical) dates in the first place, or else to construct them explicitly from the text values you have available instead of relying on the automatic interpretation.
Eike Rathke says more about this in his blog: http://erack.org/blog/archives/8-LibreOffice-date-acceptance-patterns.html .
I trust this helps.
Brian Barker