stötta is a somewhat colloquial. For more abstract meanings, like if you'd say things like 'this supports her argument', or 'I support this suggestion', we only use stödja or stöda (those two are just spelling variants). stötta is more common for emotional support
These verbs are used in many expressions and which verb is preferred can vary a bit depending on the specific expression.
as per arnauti in https://forum.duolingo.com/comment/5539306