In a situation like you're describing, where a client wants constant contact and is upset if they don't get a reply etc.—IMO the problem there isn't that the client feels attached to the therapist but really the opposite, it's anxiety about the attachment. A person who has a secure attachment doesn't need to constantly ask "are you there are you there" and isn't devastated when an answer doesn't come right away, because they know the relationship is stable without that reassurance.

ETA I don't have much contact with my T outside of sessions myself, but I believe there are some kinds of therapy that include a lot of contact (I want to say DBT but I'm not sure). So different things work for different people.