There was a similar report, and the user says they can press the Cancel button at least: https://illustrator.uservoice.com/forums/601447/suggestions/49007717
Brendan, can you?

Illustrator repeatedly gets stuck while renaming or saving files, forcing me to quit the application and redo the entire project. This has happened countless times, and the recovery file often does not work either. It is causing significant loss of time and work. This is a serious issue that needs to be properly investigated and fixed.

Stuck while renaming the file again. This is probably the 100th time I’ve faced this issue. Every time, I’m forced to quit Illustrator and redo the entire project, and most of the time the recovery file doesn’t even work.

This is a serious and unacceptable issue for professional users. I genuinely don’t understand how a basic file saving/renaming bug can continue to exist for so long without being properly fixed.

I’m paying for a professional software subscription, and repeatedly losing working time and project progress because of this is completely unacceptable. Adobe needs to take responsibility for this issue and provide a proper fix. At this point, I believe compensation for the time and work lost is justified.

I am experiencing a recurring issue in Adobe Illustrator on macOS where the “Save As” window becomes unresponsive. Once the dialog box opens, I cannot close it or proceed with saving the file. The only available action is expanding the window, and none of the other controls respond.

This issue interrupts my workflow and forces me to restart the application, risking loss of work. It appears to be a persistent bug rather than a one-time glitch.

Kindly investigate and advise on a solution or provide a fix in an upcoming update. I have attached a screenshot showing the problem for your reference.

Looking forward to your support.

Best regards,
Kavin Raj

Anytime the Save button is pressed, it opens the dialog box and immediately freezes. All of the buttons are grayed out besides the option to save to Creative Cloud itself. I cannot save to my computer or hard drive (both have tons of memory). I tried to use Illustrator 2023 and it didn't work either.
I'm running a Macbook Pro 2019 with 32gb RAM and Version 12.3.

At the moment of saving the file with "Save us" the pop up window freezes and the whole Adobe Illustrator also stops working. To be able to do something I need to "Force quit". But the action of "Save as" will not work, therefore I can not save my documents.