The “custom” began when Jews started coming in contact with Christians whose minhag was to give gifts for their winter holiday (details vary among countries – the custom goes back to the Roman holiday of “Saturnalia”, which is the origin of the holiday the goyim observe in December). A good argument can be made that giving presents at this time is clearly avodah zarah, since gift giving has always been part of the avodah of their holiday.
The “Hanukah gelt” custom involved giving a few coins to be deposited in a pushka.