I tried replicating the error you are getting & your calculated column formula seems fine.
,) does not work in formula (I am not sure but it is based on something language or regional settings on your site). So in that case use semicolon(
;) instead of comma(
Set The data type returned from this formula is: to Single line of text in column settings.
Better version of your formula:
=IF(ISBLANK([Birthday]), "", CONCATENATE(TEXT([Birthday], "MMMM"), " ", TEXT([Birthday], "DD")))
This will return blank value when there is no value in