You don't.
A backlog is part of a board - it's a list of all the issues that need some attention. The board is the same view, but limited to the items that the team has said that they are going to do, or are working on, or have recently finished. The easy description of the relationship is "a backlog is part of a board (an optional element or function)"
But, you say "each board has a unique filter" - that's exactly where I think you need to go! You probably want to create many boards, each selecting issues with a different filter, and probably with filters that are exclusive, so issues do not appear on more than one board.
The easy and obvious set of filters people might use here (and are pretty much a Jira default), are just "project = x", so you end up with board-1 looking at project-1, board-2 looking at project-2, board-3 looking at project-3 and so on.
But you are not limited to that at all. If all your issues are in one project and distinguished by a component (as @Brant Schroeder talks about), then you can have boards that do stuff like "board-1 = component-1, board-2 = component-2, etc".
You absolutely can use a custom field to do this, yes, but think carefully if you're going to do that sort of thing - some fields (including components) allow multiple entries, so issues may end up on many boards (imagine you set the "colour" of an issue to red, green and blue. It will appear on the red board, the green one and the blue one). I usually use a single-select or radio-button field to identify the team the issue should go to (I strongly equate boards with teams)
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