Yes, Beta versions are accessible via Creative Cloud Desktop app.
To apply to the Prerelease program, visit https://www.adobeprerelease.com/
This article about prerelease program is still almost relevant: https://medium.com/adobetech/how-to-join-creative-cloud-prerelease-programs-ccecc10ae7d3

How do I access the prerelease version to check it?

Hi,
I confirmed that this issue is resolved and it is working in latest version of Illustrator.
Thank you so much.

This behaviour need to change.

This has cost me so much time and embarrassment.

I'd like to give a more precise description of the problem:

I noticed that the feature works as expected when you launch "Save as" (eg. the first time you save a file) and then repeat this action before this saving process has finished. A pop-up appears that asks you whether to go on saving or to interrupt the process > see (Italian) screenshot attached.

The problem appears as soon as you save a modified file by simply using "Save" (from the menu or by keyboard shortcut) which is the most common action during a work day. Doing so the background saving wheel immediately disappears, the Save as window pops up and you are pretty close to lose a lot of your work when saving a flattened PDF as described by @anonimo.

How to recreate the bug:

When working on a large file (that takes some time to save), save the document (CTRL-S). While the document is in the process of saving, SHIFT-CTRL-S (Save as) a different filename. The initial save that was in progress will be canceled and only the save-as will complete.

When saving a file multiple times, e.g. first to save changes to a native AI file and then "save as" to another format, e.g. PDF in the current version of illustrator the initial save command is stopped once the "save as" command is issued.

In previous versions the initial save command was executed before the new save-as was initiated ensuring both files were written correctly.

Frequently with this function the initial native file save is not completed unless the full progress % in the document title is monitored by the user to be complete before the secondary save task is started.

'Save as' in background is not working properly. Clicking "Save as..." while the system is saving something in background - cancels that saving (in background) without any notice. The recently cilicked 'Save as..." should be automatically placed in queue - like in photoshop. It's a distaster it's still not solved...

In the latest Illustrator, v 25.2 on Mac, pressing Save As now cancels any Save action that is in progress... Previous versions waited until save was complete to initiate the Save As, and the last few updates have given an Alert and asked what I wanted to do... Please fix this, as this has caused me to save multiple project files as PDFs, but without the latest revisions being saved in the AI file (because pressing Save As stopped the save I had started).

While I can adjust how I save files. THIS IS A HUGE PROBLEM, AND VERY COUNTER-INTUITIVE.

If I "Save" a document with Background Saving turned on, and hit "Save As" before the process has been completed, it will stop the current "Save process and not save the file.

The problem that really needs to be fixed is there is No Warning (Message Box) that the Background Saving has been aborted and that document has not been saved.

2019 iMac - Catalina 10.15.7
This issue can be replicated on Illustrator 2020 & 2021 or 24.3 & 25.0 or whatever we are calling them now.

Yes, this! Just wrote the same exact bug yesterday, didn't notice this one.
https://illustrator.uservoice.com/forums/601447-illustrator-desktop-bugs/suggestions/41755852-file-saving-issue

Hi,

I'll give you an example:
I work with a slightly heavy multi-artboard illustrator file.
After making the neccessary design changes, I press Ctrl+S to save the .ai file.
Then I wait for a second until it starts showing the percentage of the saving progress in the "filename tab". When the percentage shows up, it means that I can -meanwhile- press Ctrl+Shift+S to also save the file as a .PDF. Everything seems fine so far - the dialogue shows up, I rename the file, press Save, select my PDF saving preferences (with Preserve AI editing ability unchecked...), and hit Save PDF. Then close the file (no dialogue pops up).

Now, when I look in the folder, I see that that the .PDF file is there and it's saved as needed, BUT, the .AI file has NOT been saved (updated)! What I'm left with is an old version of the .AI and a flattened .PDF, so I hope this issue will be sorted... This nice multitasking feature used to work properly before the update.

OS: Windows 10 Pro
AI: Illustrator Version: 25.0

While background saving a resource file (e.g. in .ai format) one needs to be very careful to not start "Save as" processes (e.g. PDF) before background saving process has finished otherwise il will be interrupted without any hint. This will lead to lose work in your resource file due to switching file format.
It would be much better like Photoshop does: background saving with alert when trying to "Save as"