Is it considered bad manners to accept a request, when it is already assigned to another artist? I accidentally clicked on a request that another artist was working on. I don't want to give the impression that we are in competition or that I think that I could do a better job. I can see how some may get that impression, and feel a little upset.

Or is it ok if multiple artists work on a request? Is it rude to take on a request that someone else is working on?

No it is not bad to accept another request as someone else has accepted it as well. Give it a go, it gives the requestor more choice, and gives you and other people a challenge, and something to beat.
JMann