PO for two different scrum teams

Is there already an owner for that product? If not, why is it being built? Who wants it and why?

The one they had left the company and now there is  no owner at all. 

Of the remaining product stakeholders, who would have the others’ confidence to best represent their interests and to maximize product value?

Are you known to those stakeholders, and would you consider yourself to be a good fit for their PO?

It doesn't seem to be a sound strategy to compromise one working product because another product is having issues. 

This is reminiscent of the old (and poor) project management resource allocation strategy, where management mistakenly believes they can simply "plug" resources into positions at a certain % of capacity (i.e. - splitting time 50/50 between two projects).

Also, it is very alarming that your management believes Product Owners are interchangeable based on their job role only, and that intimate knowledge of the product doesn't seem to be valued.

"it is very alarming that your management believes Product Owners are interchangeable based on their job role only, and that intimate knowledge of the product doesn't seem to be valued." - this sounds like jumping to conclusions, based on partial data. To me it sounds more like an effort by management to rescue a product from a difficult situation. Maybe they did the analysis and J K turned out to be the best fit? Maybe he did not, but is the most "available"? We do not have full picture.

Of course, being a PO for 2 different products might be very challenging... But as a temporary solution...

Filip, while management may indeed be trying to temporarily "patch over" a difficult product situation, my statement that his organization has a poor understanding of the PO role was based on the information provided:

• JK is a full-time product owner within the company for one product, and working as a member of one Scrum team
• He is being asked to take on product owner duties for an existing product within the company, due to a vacancy
• This "other" product is supported by a different Development Team(s) executing in a different sprint cycle
• This "other" product is supported by a different backlog, stored in a different backlog management tool

I would ask then, how could JK's management conclude that he is the best fit?   

• He isn't available (already full-time supporting another product)
• His current product support would be impacted (context-switching between two products)
• He may be facing the need to learn a new product backlog tool
• He will need to learn how to work with a new Development Team and Scrum Master (Tuckman)
• Both products would need to schedule ceremonies and meetings around his availability

Was there anyone else working with the former PO who may be familiar with that product, that Scrum Team, that backlog, and that backlog tool, and can temporarily step into that role instead of JK?

The 12th State of Agile report suggests that the top 3 of the greatest challenges facing adoption and practice are

1. Organisational culture at odds with agile values
2. General organisation resistance to change
3. Inadequate management support and sponsorship

It sounds as though JK is running into these here. Ian and Tim's responses are sane and, for us practitioners, sensible.  What we don't know however is what he has for internal company support.  That's where a truly useful response becomes problematic.  That his company have approached him suggests a low/medium maturity in the company generally so he's got an uphill struggle. As I say start with Ian and Tim's direction by proving with existing metrics/facts that splitting him would have a negative impact.