Can anyone suggest what's going on?

Browsers are free to ignore cookies, make them temporary (duration of browsing session), keep them forever even after being instructed to delete them , even decide what to do based on P3P or some random heuristic

If it works on one popular browser, chances are the correct headers are being generated, which means this is likely due to a GoogleChrome setting/plugin/whatever ... maybe even a bug

So check the headers manually, check for updates, try with a fresh profile (if chrome has such a thing like firefox does), check the settings ... I can't help with specifics i don't use chrome