How to organize Scrum

I'm not convinced that Scrum, as it's laid out in the Scrum Guide, is suitable for your organization. With 2 teams maintaining 5-6 products, there's likely to be a lack of commitment toward working with common stakeholders and focus on a unified, cohesive set of work. This doesn't mean that you can't learn from techniques used in Scrum to organize work and coordinate among teams, but I don't think that you'll get the maximum benefits of Scrum.

With your current breakdown, it's hard to achieve any of the ideal situations. You can't have a 1:1 relationship between Product Owner and products - with 5-6 products and 2 Product Owners, each Product Owner must be responsible for multiple products. With 2 teams, you can't have a 1:1 relationship between team and products - each team would be responsible for multiple products.

I would recommend reducing context switching. Making the relationship between team and product and Product Owner stable. Try to minimize differences within a Sprint so the team can be focused on one thing at a time.

You may also want to evaluate what you consider a product - you may not actually have 5 or 6 products or may be able to structure your organization in a way that would better enable your teams.

Scrum Development teams do not work on multiple products and with multiple product owners at the same time. Your points 1 and 2 are breaking the basic rules of Scrum. You are not following scrum framework for your process, therefore none of the following rules form the scrum guide have value in your process. 

I guess I had to formulate my situation more precisely.

By products I mean the projects which usually do not evolve much, once they are developed.

In most cases there is one project active for a team at a time.

By products I mean the projects which usually do not evolve much, once they are developed.

Do you have any complex emergent products for which Scrum is suited?

By products I mean the projects which usually do not evolve much, once they are developed.

In most cases there is one project active for a team at a time.

So based on those comments I want to adjust your initial comments.  I am going to cut/paste yours and bold my adjustments.

Could you please advise how to organize Scrum in the following situation:

1. 2 teams, 2 project managers

2. 5-6 projects at the same time, each project is managed by one project manager

3. two week sprints

In essence you do not have any Product Owners or Product Backlogs.  Scrum is not a framework for project management. It is a framework for incrementally delivering the correct value by inspecting and adapting as you go based on current relevant information. 

I agree with @Thomas Owens on this statement.  

You may also want to evaluate what you consider a product - you may not actually have 5 or 6 products or may be able to structure your organization in a way that would better enable your teams.

Once you are able to determine true products and establish a way to organize around those, you will be more successful with the Scrum framework.  But until then you are better off just borrowing some agile practices from various frameworks, methods and disciplines to build something that works for you.