I saw a similar problem on OSX as well and assumed we did not yet have platform-specific variants of the manual. But we might as well fix the redirects now.
Yes, as far as I know, the manual is currently identical for all platforms. I see two possible alternative solutions:
Add a patch to torbutton to send the user to the appropriate link for a given platform: https://tb-manual.torproject.org/${platform}/en-US. (I don't think tb-manual.torproject.org should be responsible for detecting the Tor Browser platform.)
Change the manual so there is a single version for all platforms on each locale: https://tb-manual.torproject.org/en-US
Both of these options are easy to implement. The disadvantage I see for (1) is complexity. Do we really need a separate manual for each platform? And if we do have a separate manual for each platform, are we going to need separate screenshot images for each platform?
I guess the main difference in usage Tor Browser platforms has to do with installation and startup. Right now, there isn't such a page, but if there were, it could (1) be different for each platform or (2) simply have a section for each platform.