Actually what it sounds like is the mom was thinking in her mind that it worked like applying for a job that you got a list of candidates, interviewed them and then picked someone for the job or in this case a child for you daycare. Maybe mom has been having trouble getting from the application to the chosen for an interview stage or up against jobs that have been filled internally by who you know. It sure sounds like it from her over reaction.
At the same time nothing you did was wrong and really would she have wanted to waste her time interviewing if there was no chance of getting the position.