Using Scrum, the Project Manager role doesn't
Correct, but the inexistence of the role doesn’t mean inexistence of responsibilities and the project management tasks. This is largely taken by the product owner. Organisations are finding these days that there is a misinterpretation to what Scrum means in terms of Project Management, and most of these organisations are bringing PMs back to do what the PO or Scrum Master are not educated or prepared to do, which is Project Management.
The Scrum Team can build a product by itself
Correct, and I would be surprised if they couldn’t. That’s the reason it exists, to build the product! PMs, Scrum Masters or PO don’t build anything.
Essentially, the Project Manager's tasks are more or less divided
between the Product Owner and the Scrum Master.
Indeed and most often these tasks are poorly performed by both.
But what happens when in addition to a product, you also have
You shouldn’t have a product without a project, regardless if you deliver it using scrum, waterfall or manuscripts in toilet paper. No product delivered by an organisation if not through a project. Ongoing operations don’t produce products, projects does.
By that I mean that a product is built through multiple projects.
Recipe for disaster. You can have individual components of the product being built by separate projects, which is called a Program (two or more projects delivered in coordination to achieve a common objective), but given the unique nature of a product of a project, it cannot be delivered by multiple projects.
For example, client A and B and are both paying for product 1, with
different deadlines and priorities of course.
Client A and B need to talk to each other and nominate ONE product owner. The nominated product owner need to manage both parties expectations and isolate the delivery team from the conflicting priorities. The PO should agree the priorities with the sponsors and then bring the list of priorities to the scrum team
Does the Product Owner of product 1 should manage the scope date and
budget of project A and B?
As mentioned before, there is no such situation where 2 projects produces the same product. You may be referring to 2 different projects and YES, the PO should pass the priorities to both scrum teams
Is it a good place to introduce someone who has the role of a Project
Manager, to manage project A?
It is essential as a the Project Manager will relief the PO from administrative tasks and allow the PO to purely focus on the product and business value.
If not, what happens when there will be too many projects to be
handled by Product Owner 1?
Product Owner need to speak with project sponsors to either bring Project Managers or re-structure the projects/product so that more POs can be nominated.
If there is a Product Owner for each project, they aren’t real PO
because they don’t have power over the product they are delivering.
Every project need to have ONE Product Owner and each project should produce a UNIQUE product. Anything different to this means organisational mess.
Another example of a similar situation that could require a different
answer is the following: Let's say a service company business model is
to build websites for clients. They build a website, ship it and then
consider that product to be done. But at some point, they decide to
build a "website builder internal SDK" to accelerate future websites
development. I'm sure we agree this "SDK product" is worthy of its own
Product Owner and whole Scrum Team. Should each service be under the
responsibility of a Product Owner that manages the dates by talking to
the "SDK" Product Owner?
Different products, different projects, POTENTIALLY different POs.