Every company that has data uses products builds on top of it. Product or service companies at least do analytics and use dashboards or reports. Some build in-house developed tools or use existing free or enterprise software, or outsource the data part. Therefore data teams play different roles, have different size and structure. Some companies focus only on data and provide tools, services or consulting. But whatever type the data team is, it always manages the data, either owned by the company or client.
Relationship between data teams and product teams are always different, but there's one thing in common. You have to balance tasks between product-related and data-related. Product-related tasks are everything that's coming from the product team - regular stuff or ad-hoc requests from dashboards to reports or complex analyses, A/B tests, some ML models and so on. Data products are for internal usage inside the team. Their aim is to improve infrastructure and data warehousing or provide the company with new data tools and frameworks.
The second type, building internal tools, is my favourite. It is always high leverage. The eventual goal is always the same - improving the way the company and the data team works with data and make the most out of it, quicker and better. Although not all internal data products are successful, it's always clear why they succeed or fail. Because there's only one client - the company itself.