This may be a misunderstanding.
It should be easily clarified or fixed. Ask you DP to ask the PILs to forward you the full booking info. If FIL is indeed listed as primary carer, check the airline’s FAQs or T&Cs to see what that means in practice. If necessary, contact the airline for advice.
If there are implications, ask if it can be changed. In fact (unless it’s a mix up and FIL meant lead booker) if he’s definitely listed as PC, I’d ask to change it anyway, unless there’s a hefty charge and it’s unlikely to cause a problem. Better to have peace of mind and clarify to the airline who the actual PC is. Could it potentially cause problems if either party had to cancel or change to another flight?
As for talking to FIL about it, it depends on (i) what he’s actually booked and (ii) the implications of that. If it needs changing, you’ll probably have to speak to him to change it, as he holds the booking. But it needn’t feel confrontational. Just matter-of-fact for good reason.
In general, I suggest your DP is clear with his DPs about speaking to him, or you both, before they go ahead with booking things for your child, as you like to consider options and decide. Not at all unreasonable. They probably think they’re saving you the stress. Lots of overbearing people think they’re being thoughtful.