I thought this was the case as well, but was corrected by Bill Hilton. The
colors on non-managed apps will be better, but not as good as in a color
managed app.

Ahhhh .. yes, you are right, it can well be that I compared
the picture that is in the Adobe RGB space - so that's why
I saw the difference.. OK, I'll repeat my tests with sRGB
picture and PSE set to sRGB space. Then the colors should
be the same in the PSE and other browsers, right?

No. Having a profiled monitor means that any application that uses
colour management "knows" what the screen looks like. That means the
application can adjust its colour output such that it can show you what
your image will look like on any other device (such as a printer)
provided the application also has access to an accurate profile for the
other device.

Your web browser doesn't take note of colour profiles. It will make no
effort to adjust its output to match PSE or any other application. It
just throws the image onto the screen. If you want to see your image the
same in Photoshop and your browser, you have to tell Photoshop that's
what you want. Then Photoshop will also "throw the image onto the
screen" without fussing over profiles. Under the full Photoshop you do
that by selecting "Monitor RGB" as your soft-proof colour space. I'm not
sure how it works under PSE.