elements on two computers

There are many options which you can use here. In addition to what Micheal already mentioned you can also consider using a cloud service. If you're using Windows then OneDrive could be a decent and free alternative (depending on the amount of data we're talking about here). OneDrive provides 5Gb worth of storage for free but you can also get a subscription which raises its capacity to 1Tb. There are some caveats to keep in mind though: although it does an excellent job for data which resides on your system drive (so: the same drive Windows is installed on) things can become tricky for other locations.

On my setup my 'Pictures' folder (which is a default for Windows) is synced with OneDrive and because of that I can easily access it on all my laptops as well. The best part is that you don't have to do anything about it...  

Store both your photo files AND your catalog on a small USB external drive.

They will be immediately accessible when you plug the drive alternatively to both computers.

The process is the same as when you migrate to a new computer:

https://helpx.adobe.com/photoshop-elements/kb/backup-restore-move-catalog-photoshop.html

- You do a full backup from the organizer to an external drive (the one you use for backups)

- Check the destination drive letter of the destination drive: it must be the same seen from both computers.

- Plug in your destination drive together with the backup drive.

- Create a master folder just under the root drive, something lile 'My Photo Library' and run the restore to that 'custom' location. Your photo file tree and your catalog folder will be restored there, and the links in the catalog will be updated to the new location. To start with that catalog copy, use the catalog manager to 'Open' it by browsing to the location of the catalog folder. Or just double click on the 'catalog.pseXXdb' file in the newly created catalog folder copy. That choice will be remembered.

- plug in the external destination drive to the second computer and do the same to open the catalog.

That's a very common process which I have been doing for years. With USB3 drives, you don't really notice a speed difference with your original setup. You may delete the original folder tree as well as the original catalog to save disk space on the original computer. Many users who don't share their library with two computers use the same process only to save space on their small SSD drive in their laptops.

Note: to delete completely your original photo folder tree and your catalog, use the Explorer, not the organizer. To delete the original catalog folder, you can use the organizer catalog manager when your active catalog is a different one, or you can find the catalog folder location from the menu Help >> System Info and delete from the Explorer. Be sure to keep the full backup created for the migration until a new one is done after everything works as expected.