I know this is a dumb question, but I am new to AE and can't figure it out. How do I open a project file without it closing the project that is currently open? Basically, I want to open a .aep file and have it show up as a new "tab" next to the composition that is currently open.
You can only have one project open at a time in After Effects.
The way you've phrased your question suggests that you may not be clear on the difference between an AE project file (project.aep) and a composition. It's the Composition(s) that you see as tabs in the timeline. One project may have one, or multiple compositions in it. If you're uncertain about this I'd strongly suggest you review a few basic tutorials at: https://helpx.adobe.com/after-effects/tutorials.html
However, you can import one project into an exisiting project. Simply by using the 'File > Import' menu option, or by double clicking in the project panel. This will import the project in it's own folder, with all the related compositions and assets. You can then double click one, or more, of the imported compositions to open them as tabs in the Timeline.
Maybe this workaraound for Windows will help:
Now every time you use the shortcut a multiple instance of AfterEffects will open. Then you can open a project in each instance..
Sorry - my english isn't the best - hope you understand
This is the actual correct answer and this is how nuke operates. Perfect for when I have to have multiple projects open at once.
You can import as many AEP files as you like and After Effects will add folders to the Project Panel that contain the assets and the comps in that project. You can open as many compositions as you like in After Effects and they will show up as tabs in the timeline panel. The Composition Panel will not have tabs by default but you can create a New Viewer, lock the view, then drag the composition panels into a tabbed stack, but I don't see the point. The comp Panel will automatically switch to the comp that is selected in the timeline.
You should spend some time in the Learn workspace to become acquainted with the UI and you should also spend a fair amount of time with the User Guide. Just poking around in the UI or searching YouTube for tutorials is an extremely inefficient way to learn how to use After Effects efficiently. You should also be very careful about the tutorials you choose. Most of the "I can't figure this out" problems I help people solve come from poorly explained tutorials with poor workflows prepared by enthusiasts with good intentions but pretty crummy recipes for the projects.