As far as I am aware, if they are not "gifted" the email account will be seen as one of your email accounts and any 2FA set up will notify you and you will have to acknowledge it every time 2FA is activated.
This would of course mean that you will have to contact the family member to ascertain if the 2FA is caused by them or if it is a genuine activation from an unauthorised person trying to access the email account.
You would be better to "gift" the email account to them. Once they have accepted the "gifted" email account and set up their own MyBT and 2FA on the account they do not need to access the MyBT account again except if they want to change the password.
They would only have to acknowledge the 2FA once and mark their device as safe. Any further activations would be sent to them and they would obviously be aware if it was activated by them or an unauthorised person and would not need to your involvement.
Easier still, assuming that you have or can get their email account password, is that you just you do all the "gifting" and setting up of their "MyBT" for them from your computer. This would allow you to mange the email account account through their MyBT and if you use your contact number for the 2FA you would manage that in the same I initially outlined above.
