I have had this with a previous T and I think about my current T everyday and imagine scenerios with her. From what I gather from anyone I have ever spoken to about this, this is completely normal. In fact my own T has written about this. I think it's in part because they don't really tell us about their lives but also because people attach to the people they are being vulnerable with.
In terms of telling her about it. I think that's up to you. I know a lot of people find it really usually, personally I feel too embarrassed and have never brought it up.