This is a known and open bug for me, reported and acknowledged unresolved by Airbnb. I have no app push notification for initial Instant Booking or Inquiry.
For all NEW Instant Bookings or Inquirys, I only get email, no app notification nor txt (I had turned on txt notifications to test if that fixed the issue, it did not).
I do get notifications of following messages of a continued conversation. Only the first, initial contact has no notification pushed to the app.
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Until this is fixed, my workaround (on IOS) is turn on email notifications on my phone for VIP only, and make my only VIP the address the initial Instant Bookings + Inquirys come in from, which is "automated" [at] airbnb [dot] com
This works for me since I do consistantly get the emails. I read here that there are variations of the bug for different people.