Reboot that's your normal but it isn't everyone's. Why does "bonding" with a baby require being away from their parents?
It is very normal not to want to be away from your baby, especially while they are small, non mobile, utterly dependant beings. If the parents are happy to do all their care 24/7 there is no benefit to anyone in making the parents unhappy by forcing a separation - there is no advantage to the baby in insisting the parent cannot be there too, so it really does sound, in that case, as if the other adult relative is treating the baby as a toy they want to themselves.
I would find that equally odd if the other adult relative did have kids and asked for a "go" with the baby, stipulating that the parent stay elsewhere. Different if the other adult relative is taking their kids to do a certain activity and offers to take a cousin of the kids' along too because that's an offer, not a request.
It is also very normal to be relaxed about being away from your baby - its just not the only "right" way to be.
I've found with all my kids that I start to feel relaxed about my kids being away from me when they reach the age where they are able to be away from me by choice - when they are mobile and choose to move away from me physically to explore the world, and when they are verbal and can express preferences and wishes of their own. That's when it makes sense to me that the bonding with other important people might well involve the relative taking the child out to do an activity they both enjoy, or sleep overs etc.