Concierge, ChiliCal, Handoff, and Chat routing all assume a triggering record is a Lead, Contact, or Account, or has a relationship to one. There's no way to trigger or route off a custom Salesforce object that stands on its own, with no lookup or master-detail relationship back to Lead, Contact, or Account.

Admins should be able to point any of these products at a custom object directly, using its own fields as routing conditions, regardless of whether that object relates back to a standard CRM record.