It means 'Don't disappear' or 'Don't go somewhere where I can't find you'. In informal language, you can sometimes add 'on me' when a person's behaviour has an effect on you, especially a negative effect. For example, if two people are having a discussion and one of them stops talking, the other person might say 'Come on. Don't go silent on me'. This means 'Don't annoy me by going silent.'

It can be taken literally or figuratively. If the father is really concerned that his daughter is going to vanish (into a crowd, run away, etc.) then he is literally telling her not to disappear or leave. But if he is saying it because he thinks she might change in some way (her attitude, if she would become quiet or withdrawn, avoid him, etc) , then he might be saying to her to stay how she is.