Walker has gone through many changes over the years. He started out as a wrestler for the super human wrestling league or whatever it was called and then he became Super Patriot with the idea of staging crimes where he would swoop in and stop them to try and outshine Rogers who he thought had become a washed up has been. Then the Government decided because they could no longer control rogers they would stripe him of the name Captain America. They took the outfit and the shield which they owned and gave it to Walker because they felt they could control him. Walker was very patriotic but he also had a really bad temper and had no problems beating the crap out of a perp. In the comics Rogers did not have super strength but Walker did. Also in the comics Cap's shield was one of a kind made out of a mixture of Adamantium and Vibranium. In the movies the shield isn't really one of a kind just rare, nor as indestructible because its only made of Vibranium. If you have access to Vibranium then you can make a duplicate shield.
In the comics Walker is given the Captain America persona and shield while Rogers goes to Stark and gets a very similar designed outfit only now with black and red colors and a new shield which is only made of vibranium, no adamantium. He then starts going by the name of just "The Captain". Meanwhile Walker who goes on missions for the government with his partner the new Bucky, a black dude who also stemmed from that wrestling federation. Bucky is much more stable and is able to keep Walker in somewhat check until they go up against a racists hate group called the watchdogs, who discovers who Captain America really is and end up killing a family member of Walkers. His sister I think. That then sends Walker over the edge where he ends up killing members of the Watch Dogs. After that and seeing Walker has become unhinged, the government starts to rethink their decision to give Walker the Captain America name.
Meanwhile Rogers ends up butting heads with Stark who has gone a bit off the rails himself with his obsession of reclaiming from anyone who has been using his technology (good guy or bad). Stark discovers the armor suits worn by the Guardsman who guard the maximum super villain prison at time known as the Vault has his technology in it so he goes and nullifies their suits which of course leaves the prison vulnerable. A prison break happens as a result of Stark's actions which The Captain has to stop. During this story The Captain and Iron Man face off and if I remember correctly Stark temporary paralyzes Rogers and takes off with the new shield he had given him.
Then soon after that, the Inferno X-Men crossover happens. The Avengers had recently disbanded thanks to the exploits of Dr. Druid and The Captain along with Reed and Sue storm of the Fantastic Four along with Gilgamesh and Thor I think form a temporary new Avengers team to handle the overflow of problems from the Inferno storyline.
By this point Walker has basically gone rogue and Rogers sets out to stop him, the two go at it and eventually it is discovered that the Red Skull who at this point was thought dead was really pulling all the strings that had caused Rogers to be striped of the Captain America name and given to Walker. Redskull had discovered away to clone Rogers body and place his consciousness into it giving him a new young vital body. He then assumed the identity of a Congressman where he was able to put his evil plans in motion. By the time Rogers and Walker discover it was Red Skull who was behind everything that had happened to them, it was seemingly to late. After Walker is incapacitated, Skull and Rogers fight it out and Skull is about to blow red smoke from his cigarette that would disfigure Rogers face when Walker manages to muster up enough strength to throw the Captain America shield at Skull which causes Skull to bite down on his own cigarette which in turn cause the red smoke meant for Rogers to blow back on him disfiguring his own perfect face so that it looks like an actual Red Skull. Up until this point the Red Skull was actually just a mask. Red Skull escapes to plot another day.
Walker having saved Rogers life and both realizing they had been played by Skull all this time come to a temporary truce and Walker who realizes Rogers is a true hero gives him back the Captain America persona. The government also agrees to allow Rogers to return as Cap being they had allowed Skull to infiltrate their ranks.
Walker gets some therapy, temperately changes his name to Jack Daniels is given The Captain outfit and a vibranium shield of his own and renamed USAgent. Shortly after that the Government forces the West Coast Avengers to take him on as a team member so he can watch over them and where he ends up butting heads often with the then team leader Hawkeye.
West Coast Avengers then come to an end, Walker throws his uniform and shield out into the ocean only later to get a totally revamped outfit and energy shield so he can join the reformed West Coast Avengers team created by Tony Stark now named Force Works. Force Works is short lived though so Walker disappears for awhile until Marvel decided to give him a limited series where he has magically returned to his original The Captain outfit and now has a bird shaped shield and is watching over a prison.
He then turns up again in the short-lived Alpha Flight reboot with former West Coast Avengers/Force Works teammate Spider-Woman (the second one).
He then once again turns up in Thunderbolts where he is now the new warden of The Raft. There he is paralyzed by Nuke and left to be in a wheelchair.
His legs are returned to him in a Dark Avengers story where he and others are transported to an alternate reality for a while. The last time he was seen in the comics I believe was the Secret Empire storyline which I didnt read.
I probably missed a few other appearances here or there but that gives you the general gist of it.
Anyway I don't know if they will say the Government owns the shield in the show, but they def could claim they own the name and costume and it wouldnt be two difficult for the Government to replicate a vibranium shield. Though technically they could say Howard Stark at the time of making the original shield was working for the Government and hence it in fact does belong to them and being Tony Stark is no longer alive there would be no one else to claim ownership of it.
Keep in mind Wilson and Barnes may still be considered fugitives of the federal government after the events of Civil War and may be looking for them to answer for their crimes or at the very least not keen on the idea of either of them being the next Captain America.