In our team all task's steps are always done by the same developer. I.e for task 1, person A does design, development, and testing; for task 2, person B does does design, development, and testing, etc. Does it make sense to use Kanban in this case?
I would suggest still using Kanban by slightly changing the process:
Putting all tasks with 1 developer as Chad pointed out is a recipe for disaster while not in the initial stage but most certainly down the road when maintenance on the project would be required.