There could be a few answers here, so I'll try to hit each:
First, does work have to go through you for your process to work? Especially with newer teams, having all work flow into the process through a team lead or manager can be helpful to make sure people aren't being distracted and the prioritization process isn't being side-stepped. As the team and the organization gets used to how they work together, that person's role may become less of a gate keeper. That can be uncomfortable at first, but ask if there is any pain being caused - is the team member getting distracted or are people slipping their request ahead of others that should be first?
Second, you said that people are assigning work to a teammate. This does seems problematic. Work should go to the team and then the appropriate person should take it. Again, the details are important. If they're placing an item in the queue and your team mate says "Oh, I'm about to be done with this task and it looks like you're next, let's chat." then that's pretty ok. If they're short-circuiting the process, you're right to raise the concern and your teammate should also be directing them to the proper process.
Which brings us to the final part: is it clear to people outside of the team how work gets brought to your team and prioritized. In the past, I've worked in teams where we thought this was clear - you entered a ticket, it went in a queue, we worked top-down. Turned out almost no one knew that. They just thought they entered a ticket and we'd get back to them when we got back to them. Ask the people going directly to your teammate why they are. There may just be a small visibility problem that'll clear everything up.