Is it typical for a person with Lyme disease to have symmetrical joint pain on both sides?

Yes. Lyme disease can cause symmetric polyarthralgias - pain in numerous joints on both sides of the body. Although rare, lyme disease is usually easy to treat if identifed quickly. There are blood tests so if you and your dr have a concern, it shoud be pretty quick to rule in/out.
Not typical. In lyme disease, joint pain typically occurs first in one joint, then if not treated, can migrate to involve other joints, randomly. It is conceivable but not typical that in this fashion the same joint in each limb is involved.