On-Premises installation of TFS 2015 I am able to achieve this sort of configuration:
I was able to achieve this by updating the "PortfolioBacklogs" section of the ProcessConfiguration.xml file.
The only way I was able to get TFS to accept this configuration was by configuring the "Monthly Patching Requirements" Portfolio Backlog item as Child to the "Technical Work Request" Portfolio Backlog item using the "Parent" attribute like this:
<PortfolioBacklog category="Microsoft.MonthlyPatchingRequirementCategory" parent="Microsoft.TechnicalWorkRequestCategory" pluralName="Monthly Patching Requirements" singularName="Monthly Patching Requirement" workItemCountLimit="1000">
These items should not be related like this or in any way at all really. I want them both to be "top-level" backlog items and have all work items for both categories show up in the project Backlog and on the board. Is it possible to configure in such a way?
No this is not possible. The backlogs are always in a hierarchy and you need to put them at a specific level in that hierarchy.
The best way to handle this is probably to add a new work item type and put it in the Requirement category. Then configure the Board view to have a separate swimlane for your Monthly patching requirements.
In recent versions, a new filter bar was added which then quickly allow you to filter for the work item type you're after. It may be worth an upgrade in your case.
Conceptually the backlog contains all the types of work so that a single order can be accomplished, having multiple backlogs at the same planning level is against how the tool expects you to work.
Some alternative options you could consider: