Why don't we stutter around family members as much as strangers? isit due to familiarity?

You may be right: Especially in your case, since you have social anxiety, this makes very good sense. It's possible that a good CBT therapist could coach you to learn better control over it. Some people w/ stutters are different and have it more severely w/ family. For some it is not related at all to anxiety. We're all unique - and it seems you may know yourself pretty well. That's good and would help in therapy.