I'm interested in understanding how agile project management differs between a single product-focused team and a digital agency or software development consulting firm.
For starters, in a situation where the consulting firm or agency has multiple clients, is the Client the same as the Product Owner?
Represent the Stakeholders to the Agile Team
The primary role of a product owner is to represent the needs and desires of the stakeholder community to an agile delivery team:
Represent the Agile Team to the Stakeholders
A secondary role for a product owner is to represent the work of the agile team to the stakeholder community:
Whoever can do these well will be a good Product Owner. As CodeGnome pointed out, to be effective, the Product Owner should be an integral part of the Scrum team. We tried your approach of a client (in our case an internal client) playing the role of a Product Owner part time. It was not that effective. When we brought in a person to be a full-time Product Owner, the change was dramatic.
[I]n a situation where the consulting firm or agency has multiple clients, is the Client the same as the Product Owner?
No, they are fundamentally different roles. While the Product Owner may come from the client, the Product Owner is a specific one-person role within the Scrum framework. The client is a stakeholder (a "chicken"), while the Product Owner is a formal member of the Scrum team (a "pig").
Note: See The Chicken and the Pig for more on the use of this terminology in Scrum.
The client is, perhaps axiomatically, a stakeholder in the process. An individual or steering committee within the client organization is designated as the project sponsor, and of course the client is responsible for funding the project. However, Scrum assigns the Product Owner role as a proxy for all the stakeholders in regards to prioritization and scheduling.
The Product Owner works with the stakeholders to create the Product Backlog, and to build stakeholder consensus for project priorities and resource allocations. As a defined role in the Scrum process, the Product Owner must be an active participant in the project, and not simply a stakeholder waiting for features to be delivered.
In practice, the Product Owner needs to be an integral part of the Scrum team. As such, it is generally beneficial for the Product Owner to be a member of the same organization as the Scrum Master and the developers. In this way, the stakeholder proxy role is properly situated inside the project.
Larger clients often have their own PMO or project management framework; such clients may provide their own internal project/product manager (or other stakeholder proxy) to each project. In such cases, the two proxies coordinate with each other: the client's internal proxy represents the client to the Product Owner, and the Product Owner facilitates the client's interests within the Scrum team.
For each product, there is one person who cares enough about that product to champion it and either provide the budget to develop it or arrange for the budget to be provided.
That person is the Product Owner (occasionally it's a team rather than a single person).
Everyone else, whether internal or external to the agency, is a proxy for that Product Owner, regardless of any roles or titles that may be awarded to them.
In addition to the Product Owner, there are likely to be stakeholders whose needs have to be met before the product can go live. Users are a great example of this - they won't like a product that's hard to use - but it may also include advertisers, auditors, legal staff, architects, etc. In this situation sometimes a proxy for the Product Owner may be more effective than the PO himself, as they may have a greater awareness of these "gatekeeping" stakeholders. I would expect that to be the case in any software consultancy which specialises in a particular domain - for example, Telecoms, Commodity Trading, Banking, Retail or hotels will each have their own set of well-known stakeholders.
It's likely that PO will be a client or within the staff of the client, and so will some of the gatekeeping stakeholders, but not all of them.
