Yes. That setup is preventing all of your devices from syncing.
iOS devices allow you some flexibility regarding what Apple IDs you use with what services. For example, you can have a different Apple ID for iMessaging, and a different one for iTunes purchases, and yet another one for iCloud. The problem comes from iCloud as anything that is related to iCloud can only sync across 1 Apple ID. It just so happens that Photostream, and Safari bookmarks are two iCloud services.
The problem is that Contacts and Calendars (less so) are often synced through iCloud, as well as Reminders and Notes. Also, her having another iCloud account means she gets 5GB free storage and you two don't have to share that.
So, you will have to decide what functionality you want to give up to gain different functionality. If Photostream is important to have consistent across all devices (or Safari bookmarks), then you will have to do what you thought - delete her iCloud account and add yours. Note, you don't have to delete her account any more then select "Delete account" on her iPhone and iPad. Generally speaking, this just signs out of the account.
Unfortunately, like you thought, there is not a way to have two iCloud accounts signed into on one devices.
Personally, I wish Apple would develop a "Shared" Apple ID system which would allow one to share certain services such as Photostream, yet keep Mail, Contacts, and Reminders separate.
Best Answer
You can group the photos by the camera that took them, using Smart Albums.
First click on one of your girlfriend's photos and Show Info under the Window menu. You will see something like this. It's the camera model that is important (I have boxed it in red).
My hope is that your iPhone is not the same model as your girlfriend's. If it isn't we are in business. Assuming that they are different here is what you need to do.
Go to File/New Smart Album...
Call it "My Girlfriend's Photos" or whatever (I called mine "iPhone 6s pictures"). Set the Camera Model to include what you saw in the Info pane. In my case, it's iPhone 6s. See below.
If you show the Sidebar (View menu) you will then see your Smart Album somewhere in it. Here's how it looked here.
If you are showing the Sidebar, and you click on "All Photos" under the heading "Albums" (not "Photos" at the very top), you will notice that View/Sort will say something like "Keep sorted by oldest date added." This may give you a second chance at figuring out which photos are your girlfriend's, because the sorting is by date ADDED and not date "taken." So the pictures that were newly added would be at the bottom, and those are probably your girlfriend's. You could select them and put them into a regular album and you'd be all set.
Personally I am hoping that you two have different iPhone models because the Smart Folder method would be very cool.