I assumed without checking that I could switch between two accounts in the Element desktop app on Linux. This turned out to not be the case. However, the Ubuntu Community posted a useful guide detailing several workarounds. I tested the first workaround. Assuming you are using the Element desktop client, open Element with your already-established profile. Then go into the console and run:

element-desktop --profile account2

This will open up a second instance of Element where you can sign into a second account or create a new account. If you want to use your second account, simply launch it from the console using the above command. This is a neat solution. I tip my hat to Ubuntu Community for the very clear and concise guide.