Background: A small tech company has just grown from a flat startup team of 8 developers with no real structure to ca. 40 people (roughly 20 devs + 20 various business and management positions), who need to work in teams and have clearly defined resources and priorities. The company uses agile terms (like a Product Owner and Backlog), but few people have read more about it than the Agile Manifesto. A position of Agile Coach is open, but nobody is hired for it yet.
Situation: The CTO pushes forward a product-oriented structure for the whole company. While the devs had no problem defining what their 3 Products are and how to deliver them, the management used a literal approach: "If a Product is anything with a backlog and features, and a backlog is a list of my tasks, then my tasks are my Product."
As a result, the marketing team has a product "Company webpage" and a product "Brand development", while the office manager has 3 products: "Office management in City A", "Office management in City B", and "Travel Booking."
Question: Are there any dangers or side effects of defining company management as a set of products?
If no, how to organize this work in the most effective way? And if yes, how to define what could and could not be a product?