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Usually date format issues are national differences (e.g. mm/dd/yyyy in the US and dd/mm/yyyy in the UK), but the "reverse" date format of YYYY-MM-DD is a neutral ISO format for date representation and is really intended for internal computing use. The advantage is that it sorts correctly when stored as a number or string (since year is more important than month, which is in turn more important that day).
FME (on which the Universal Translator is based) handles date fields quite well, so I wonder if your translation is writing a TAB file with the date field as a string, rather than date, type. In the absence of any directive to convert into a particular format, date-to-string may well output this way. If this is the case, I don't know why a date-type field is not being used. Perhaps check that your source SHP is really using date fields and check your output TAB for the same.
I've just tried a SHP -> TAB translation with the UT using some data with multiple date fields and things seem to convert OK. It's therefore a puzzle as to why you're seeing the ISO format. It would be worth finding out if it's actually the data itself or the MapInfo/PC locale settings which result in dates being shown this way.