I got a solution.
Instead of changing the propertyToSearch. I simply changed the code behind to list the values based on the input provided.
So, if month is selected. I pass a parameter "searchProperty" in url as month and if day has been selected I pass a parameter as day based on the searchProperty value. I list the values and pass it on the dropdown list.
$("#demo").tokenInput("../Management/getDaysAndMonths?searchProperty="+$('input[name="suggest"]').val(),
{
//code
});