Hi
A priori, there isn't any way to convert a game purchased on Steam into an actual "account" for Amanita, with download links for all versions (including Linux one) and soundtrack. This is the major issue with Steam, you get less things and you depend on a only store which works on a only system.
However, it seems that the executable in the Linux Demo version could be the same than in the complete Linux version. So, try to
download the demo, to take all the 00-11 folders from your Steam installed Machinarium, and to put them instead of the equivalent folders of the demo. It could work.
Note than the Linux version is not a Debian package (.deb) but a compressed archive that you can extract anywhere to play directly. And if the folder replacement method doesn't work, you can take advantage of
the current promotion (all Machinarium + Samorost2 + their soundtracks for $10 instead of $25), before tomorrow!