It looks like version 26.0.3 does a colorconversion from AdobeRGB to sRGB when saving for web, but is doesn't save the profile in the png-file. Saving as PNG and Saving for Screens does not colorconvert, and does not embed a profile. A webbrowser like Safari is capable of reading embedded profiles, but doesn't find one in the PNG that is created in Illustrator. Each situation is bad, so you get the wrong colors in webbrowsers when starting off with something else then sRGB in Illustrator. But nowadays webbrowsers are capable of reading and respecting embedded profiles. And P3 is going to be the new sRGB and has a wider gamut. So why should we stick with the old sRGB while our environment has changed and webbrowsers are smart enough to read embedded profiles? The only solution seems to me that programmers at Adobe should improve the way Illustrator works. That means: any png or jpg that is being exported should have the current document RGBprofile embedded in it. On top of that it should be a users choice to convert to sRGB, and have that embedded as a profile. Just like what Photoshop does. And when you're at it: the same goes for InDesign.
And as Ton says: when placing an image with an embedded profile, Illustrator completely ignores the profiles. This is so unprofessional!
These problems are there for many years already without Adobe doing anything to solve it.

This is a must!
Whats the point to have export for screens in Illustrator,
when most graphic design is created in cmyk and Ai does not offer an option to convert to SRGB just like Ps does?
It renders it useless, as cmyk jpgs appear with inverted colors.

My vote would be to :
1. Always embed the document profile.
2. Offer a checkbox to override and convert to sRGB and Embed the sRGB profile instead.

A professional app should not leave users guessing the color space of an exported document.

Dave

And that needs to change! It should save the profile in the image, if one has used that profile in the AI file

Actually, Illustrator uses the current RGB profile to calculate the exported colors, but does not save the ICC profile with the image.
When opening or placing a PNG file Illustrator ignores the profile even if it is available.
Edit: currently (2024) Illustrator does read embedded profiles...

I discovered most people don't notice this happens because they assume it's a function of the screens they test on being different or that they simply don't know this is happening behind the scenes. I was the latter for awhile. It would be great to have the ability to embed ICC profiles for PNGs in Illustrator and Indesign instead of having to jump to photoshop. I understand the apps in the Creative Cloud work in sync but this adds too much of a cumbersome extra step to a process you've already simplified really well with the export for screens feature.

See also:
https://illustrator.uservoice.com/forums/333657-illustrator-feature-requests/suggestions/39521494-png-export-embeds-the-wrong-icc-profile

Yeah @Ton that's the case so it seems. I've been waiting for this update for so long. PS has this. Adobe owns both the software yet the bridge in capabilities is not there. The software has grown leaps and bound over the years. Adobe are you listening?? Please pitch in.

It seems that Illustrator uses the RGB Working space profile for export to PNG, but does not embed that profile.

I'm voting for what I feel is the more import part of this idea, the conversion of CMYK to SRGB. The default screen profile for the web and Microsoft Office is SRGB. At the very least, there should be a checkbox to export to SRGB. And a checkbox to include a profile.

The old Save for web (legacy) export did this automatically. It also had the option to embed a profile.

Why is it not possible to export/save an illustration as a png image with the sRGB profile included just the same way as it is possible for jpg images? The difference is obvious when viewing the png image in Firefox (in Safari and Chrome the colors appear correctly).
 
The easiest workaround for this problem that I could find is to produce a png image with illustrator and assign the sRGB profile afterwards in Apple Preview.
Is there another way to achieve this result  in one step within Illustrator? Just checking the 'convert to sRGB' option does not seem to make any difference.

I see a difference in options between photoshop and illustrator. For Illustrator, it would make sense to export for screens and force a conversion to sRGB –even when your drawing is still in CMYK. I notice that Illustrator doesn't change colormode, while Photoshop does (when exporting, PS converts to the working RGB or sRGB if you make that choice).
Besides that, I think it is very important to add compression to PNG files. And why not embed colorsyncprofiles in PNG files? It's a benefit when profiles are embedded because webbrowsers will show the image as similar.

Bumping this to see if it gets some support as this feature is still not available.

Please offer this feature for the next update! We're working with web-designers to whom we have to give png and/or eps files and if there's a specific colour profile added in the Illustrator file, it does not carry forward to exported png/eps file. Photoshop has that feature, export to pdf from illustrator has that feature. But png and eps don't please include this feature.

From Photoshop you can choose to embed a colorsynprofile when saving or exporting to png. I need that in Illustrator as well. Webbrowsers are smart enough to recognise icc-profiles and adjust display appropriately. Illustrator needs to have this option as well, and save the currently assigned rgb-profile when exporting