Remington actually made the A5, with Browning markings, at two periods, from 1905-1910, and from 1940-1947, when the Belgian factory was out of business. But those guns were not marked as Model 11.
Browning was what we would now call a "free lance" designer. He had no factory and never entered series production for any of his guns. He sold the rights to his designs to whichever gun companies offered the best deal.
For a while, that was Winchester, then Colt, then FN, then Remington. He sold rights and also worked on a royalty basis. The neat part is that he made a lot of money, but never had to raise capital, build buildings, or buy production machinery. He made drawings and models until he got the idea working, then turned it over to someone else to worry about.
Companies were so eager to buy a Browning design that they even did all the patent paperwork for him. And in one case, that rose up to bite them. Winchester's patent attorneys drew up the patents for the A5, and made them very tight. But when negotiations over a purchase vs. royalty agreement broke down, Browning took his design to Remington (the death of its president broke up that deal) then to FN. When Winchester decided to produce its own autoloading shotgun, its chief designer, T.C. Johnson, had to work around the tight patents his own company had drawn up.