Who will be the recipient of the chat from the page?

All admins will receive notice of the new message and anyone can respond to it.

Is there any to trickle it down to just one admin?
or is it facebook thing?

Thanks for the swift response

This is a facebook thing. You need to have the fanpage inbox open in your browser to respond to messages, so not all admins have to have the fanpage inbox open.

Thanks once again but the current issue is that its not directing the messenger app on mobile

its working on some mobile devices had to keep testing it on many.
guess the issue is not from your end.
thanks once again

Thanks