The reason why you're unable to redeem the gift card is that gift cards cannot be used with subscriptions such as Premium Student or Premium Family. This is so because these 2 subscriptions are already with a discounted price.
There are several options you can use if you're the manager of the Premium Family:
Give the gift card to someone who has Spotify Premium
Cancel your subscription, wait until your Premium time is over and your account is reverted to Free. Redeem the card and only after that upgrade to Premium Family again.
Note: Then you will have to invite the members of your plan again
We hope this helps! Keep us posted :)
Help others find this answer and click "Accept as Solution".
If you appreciate my answer, maybe give me a Like.