Yes, this can be set up so that every enquiry that reaches you is assigned to one named person as soon as it arrives, and nobody has to decide who picks it up. Instead of a shared inbox that everyone looks at and nobody owns, the enquiry lands in the system, a rule decides whose it is, and that person gets the alert. The rest of the team can still see it, but they can see it already belongs to someone.
The rules are usually built around the way you already work. You can hand enquiries out in turn so the load is shared evenly, or send them by destination so the person who knows that region gets it, or by language, or by whether it came from a direct customer or one of your sub agents. You can also say what happens when the owner does not touch it after a set time, so it moves to someone else or to a manager rather than sitting there quietly. Holidays and off days can be taken into account so enquiries do not go to someone who is not at their desk.
The honest limit is that the system can only share out what actually reaches it. Enquiries that come through your website forms, your booking engine or a connected chat or mail box can be routed. Ones that arrive as a phone call, a personal message or a message on a channel that is not connected have to be entered by your staff first, and whether a particular channel can be connected depends on your plan and on what that provider allows.
The next thing to look at is a list of every way an enquiry reaches you today, and which of those you would want shared out automatically, because that list is what the routing rules get built from.