In the nether, I have a 3x3 portal. The left side takes me to my base, as I would expect. The right side, however, leads to a completely different portal. Is there a reason it's acting this way?
Inside of the game's coding, the nether portal itself is merely a type of block, and the different blocks of the same portal are only connected inasmuch as they were created by the same ring of obsidian.
A nether portal takes you to the nearest available portal in the other dimension (or the one it was originally tied to when it was created), and if there is not one available, one is made. This is tracked by block, and not by portal, so in rare situations, it can create the effect you describe.
A Nether portal's target location is determined by the closest portal to the coordinates of the portal converted to the other dimension. However, portal blocks do not have any affinity to the portal they are in and follow the same rules.
For example, if you have a portal in the overworld which streches from 7, 80, 0 to 9, 83, 0 then when you go through on the left side your coordinates will be converted to a different nether position. In this case, X 0, Z 0 vs X 1, 83, Z 0. The closest portal in the other dimension is found using the pythagorean theorem searching a 256 by 256 by 256 area. If you are in the overworld, make sure your portal doesn't have one side in a different chunk to the other and that should fix the problem. If you are in the nether then it may be harder. You can try and get rid of surrounding nether portals but if that is not an option you could try moving the portal to different coordinates where the closest portal is a different one. Another clever solution could be to move the overworld portals in the y coordinates so they are separated. Or make a redstone system to turn on and off the portals when you go through them to stop interference. You can do this by putting a dispenser which drops lava and takes it out quickly so then it will not be reachable but if you go to it, you can light it up with flint & steel.