Could we say that the product owner is in fact a business/functional analyst in Scrum?
No. While the Product Owner may perform analytical tasks, the role of Product Owner is significantly broader and more central to the process than the typical business analyst role.
Product Owner (PO) Role vs. Business Analyst (BA) Role
Among other things, the Product Owner "negotiates priorities, scope, funding, and schedule[.]" Compare this to less authoritative role of the Business Analyst, which typically will "recommend solutions that enable the organization to achieve its goals."
The Product Owner often has budgetary authority in the form of Product Backlog prioritization, and the role's ability to set goals and priorities for a project are generally much broader than the typical analyst's role of defining or refining specifications.
Some Scrum teams may actually have business analysts as members of the team. On other teams, the business analysts may be extrinsic to the team, but work with the Product Owner or the Development Team as an interface to (or proxy for) the actual end customer.
While you could technically appoint a BA to the Product Owner role, it would be a mistake to consider the two roles as functionally equivalent. In general, a Product Manager or Executive Sponsor would be a more comparable job title for the person assigned to the PO role. However, keep in mind that job titles (and job responsibilities) outside of Scrum's well-defined framework may vary widely between organizations, so your mileage in drawing comparisons may vary.