When talking to my kids about my sister, most times I use "Aunt," but every once in a while, I'll just use her name.
My parents always referred to my godmother as "Aunt," even though she, like Lily, Robin, and Marshall, is not related to me by blood. Again, my parents will often just refer to her by her first name.
And we have always been given the impression that Ted's story is being told continuously. It would be more odd to drop the "Aunt" in the same telling of the story in which he used it as a sign of things to come, rather than an honest omission.