I would argue that Jackson was a democrat for two reasons:
First, many of Jackson's actions were actions that the common people (as opposed to the elites) would have wanted him to take. This includes, for example, his efforts to kill the Bank of the United States and his "removal" of the Indians from the Southeast. Doing things that the majority wants is something that we would typically call democratic.
Second, Jackson was a symbol of democracy because of his background. He was the first person from a (very) humble background to become president. This was symbolically very important.
Since Jackson symbolized democracy and because he did a lot of things that were popular with the common people, I would argue that he was a democrat.