See or count the amount of emails that a specific Microsoft Exchange mailbox receives

As already mentioned, the way to go is with a ticketing system of some kind, but if that's unacceptable for some crazy reason, the best solution is going to be having a "ticket clerk."

Someone whose function is to monitor the incoming tickets (or emails in your case) and assign them appropriately.

In environments where there aren't enough tickets to justify having a full time position for this, it's often done as a part of someone's job function (like a Tier 1 desktop tech or an IT manager), or via rotation (today's it's Joe's turn to monitor the queue, in addition to his regular duties).

You'd have the "ticket clerk" as the guy who's in that shared inbox assigning emails to the group to take action on, while the rest of the group would stay out of it, to the extent that's feasible. In your shared mailbox, for example, you could have a process where the ticket clerk would "assign" an email to a technician by forwarding it to the technician, and then moved from the inbox into a folder like "Assigned to Joe," for a very simple implementation.

Technet has all the information you could want: Reference

Depending on your preference for a GUI or PowerShell you can follow the instructions to create a new discovery search against a mailbox or against the entire organization. You can put the items on in-place hold if needed. With your search results in hand, you can then export the items to a PST or another mailbox (Discovery mailbox, typically).

Generally speaking you will want to review the search results for validity before exporting.