It's a bit complex, and I can only speak to the US and England.

In England, up to 1715, muskets were typically bought by the regimental colonels ( who had often bought their positions) for their own units. This meant that there was not only variation in caliber, but variation in quality and a chance for corruption: the government would pay the colonel for good muskets, the officer could buy bad ones for his soldiers and pocket the difference. In 1715 there was a reform; the New Land design was chosen to be a standard model, and by 1722 there was a system. The government contracted with some very big gunsmith shops in Birmingham to supply the metal pieces, like locks and barrels. The barrels were proofed and the pieces stockpiled in the Tower of London. Contractor gunsmiths were supplied with the metal pieces and a model of the musket , which they were expected to reproduce. That New Land Pattern became known as the Brown Bess. Smaller numbers of specialty guns might be contracted and given the names of the makers- therefore the rifles made for the army by Ezekiel Baker in 1800 are known as Baker rifles. That system of government contracting did really not work very well during the Napoleonic Wars, and a government armory , the Royal Small Arms Factory was established at Enfield: thus, the 1855 Enfield rifle, the Lee-Enfield, Short Model Lee-Enfield, etc. That armory would continue (though with big changes) until the final privatization under Margaret Thatcher in 1984 ended it.

In the US, there were two armories established at the end of the 18th c., one in Springfield MA and one in Harper's Ferry VA ( now WV). Those armories were arms factories and, like Enfield, would produce their own models- therefore the 1855 Springfield Musket, the 1803 Harper's Ferry Rifle.

The armory system in the US eventually had production problems: in WWI, the Springfield Armory could not produce enough of the complex 1903 Springfield Rifle to outfit the Army, and the government had to contract with makers like Remington to make the simpler P-17 to fill the need. Contractors like Remington and Smith Corona would also make rifles for WWII, and it was during the Vietnam War era in 1968 that Defense Secretary ( and previous business executive) Robert McNamara decided that the Springfield Armory was a pointless holdover from the previous century. The US Army stopped making its own guns, and started to simply award contracts for them: just like it buys uniforms and tanks.

In the history of the Armories, there sometimes- but not always- would be an acknowledgement of the designer of the gun. The Lee of the Lee-Enfield was James Paris Lee, the Garand of the M-1 Garand was John Garand. But the name would not always reflect the most important designer if there were multiple ones; the British Martini-Henry was based on a system by Henry Peabody, and after improvements by Swiss armorer Friedrich von Martini was known initially as the Peabody-Martini; but as soon as a different rifling by Alexander Henry was added, Peabody's name was removed . Certainly having a name credit didn't mean profit for the designer, though: the British would adopt Jacob Snider's conversion for their Enfield rifles in 1860, which became known as the Snider- Enfield, but avoided paying him...and he died in poverty. Somewhat luckier was Springfield armorer Erskine Allin, who both created the breech-loader conversion for the Springfield 1855 rifle and mostly guided the design of the Springfield rifle that was based on it. The conversion was known as the Allin Conversion; the rifle that followed was known as the Trapdoor Springfield. He may not have become rich or gotten full credit, but he got some credit: and had a very long career in a government job.

The Springfield Armory is now a National Historic Park, and though the site is quite small compared to its glory days it's well worth a visit.