Just for a test, try a different browser.
I can confirm that "save to Library" currently works for me ... but on Windows 10 and Firefox.
The same issue showing up in both Firefox & Safari.
We noticed that this issue is due to some libraries having an empty name. To solve the problem, you can go on assets.adobe.com, go into Libraries menu and rename all the libraries having an empty name. Once done, please refresh your Adobe Stock page and you should be able to save again into your Libraries.
Please note that our tech team is working on a solution to properly display empty names libraries on Adobe Stock as it is made on assets.adobe.com.
I have at least six which are named. If you watch the clip you'll see it doesn't give options to save to any.
I watched the clip and can see that your situation is similar to other customers for which we solved the issue by renaming all the libraries having an empty name through https://assets.adobe.com/libraries.
Please go on assets.adobe.com and rename all your libraries having an empty name. Even if most of your libraries have a name, having just one unnamed library is causing the issue you are seeing on Adobe Stock.
Many thanks, this has indeed solved the issue.