I guess I'm wondering if there's companies with their own specific websites who specialize in making game assets and selling them.
Not many, this isn't very profitable. Most artist will sell some stuff on popular sites or on their blogs.
A artist isn't some kind of on demand asset store, all of us know how to both make and implement our art.
If you ask a artist for a character that can run and jump, they will make the sprite, program the running and the jumping and place it in the game for you. Because how the character moves and responds, is part of the art.
So getting to the point, all assets on the asset store is defective products:
We sell them knowing that they wont work as intended and the buyer knows they have to modify the art to work with there game; they will be doing half of the artist's work. They also have to share, and take the risk that it doesn't work.
Assets takes a very long time to make, selling on a market doesn't guarantee enough sales.
No artist wants to focus a huge amount of time making something that won't work as it should, so most market assets is low quality or just some random piece a artist made as training.
A artist also can't charge full price for a market asset because of the downsides, as such the unspoken rule is to sell at 10% of the usual worth.
Most developers are amateurs when it comes to art, so even if a artist spend days making a good piece, it will only look cluttered to a inexperienced user; so most art on market stores are basic low quality stuff.
However the most damming part of market assets is the fact that instead of expanding the art to work with the game, you will have to change the game so that it can make use of the limited functionality of the market art. (Unity games is a good example of this.)
For these reason the demand for market assets are low, a new developer will use them a few times ,then move to using a artist when they can afford it.
A artist makes around the same amount of money, from one asset in 2-3 months on a asset store as they would if sold to a customer. They could also sell the full intellectual rights for more than double the amount a asset would make on a market.
So in short, making market assets is less profitable to an artist.