"First FN Browning was the 1899 .32, revised only a little to become the long running 1900.
First Colt Browning was the 1900 .38."
Browning started working with FN in 1895 or 1896. As far as I can tell, he didn't start working with Colt until 1898.
FN originally wasn't interested in the larger, more powerful Browning designs, but did come around.
Around the same time Colt and FN entered into an agreement whereby Colt would make Browning designs for sale in North and South America, and FN would sell the same guns under their mark in Europe and Africa.
Browning ended up selling the same designs to both companies under the agreement.
And, it wasn't just pistols, either. Shotguns and rifles, as well. A few years ago I saw a Remington Model 8 in .35 Remington at a gunshow.
Only it had been manufactured by FN and was stamped 9mm...