As the name suggests, the PRODUCT Owner is the owner of the single product. The product owner focuses on the given product by constantly being in touch with the customers and through the expertise in the market understanding. Aligning a PO for multiple projects will IMPACT the quality of deliverable and it will also affect the individual playing the role of a PO. 

The focus gets divided, the time also gets broken down between different parties which in turn creates a mess for the product owner. It is not advisable to align ONE product owner with multiple projects as it also affects the strategy and timeline for the project. It is like asking an author to write two books at the same time, it is difficult to justify the efforts and we end up with chaos. It is difficult to multi-task, manage multiple stakeholders, manage his/her throughput of deliverables across products, prioritize the tasks across product TEAMS, etc. Though few of the organizations are aligning their Product Owners to more than one product, again, it then depends on their ability to DELIVER the right thing.