Technically LR can be loaded onto both a Windows and Mac system using the same serial number, since it does not require activation. The primary stipulation is that you cannot use the program simultaneously on two systems.
LR 4 EULA:
Portable or Home Computer Use. Subject to the restrictions set forth in Section 2.1.4, the primary user of the Computer on which the Software is installed under Section 2.1 ("Primary User") may install a second copy of the Software for his or her exclusive use on either a portable Computer or a Computer located at his or her home, provided that the Software on the portable or home Computer is not used at the same time as the Software on the primary Computer.
I'm not an attorney, but I see no restrictions anywhere in the LR4 specific EULA that apply to your usage.
Thank you very much for your valuable feedback.
One further (technical) question regarding this topic: Is there a recommended way to use two different devices with the same catalog? Like... "I import photos 'on the road' into my laptop, start to adjust them (white balance, filters, other stuff,...) and at home I'd like to merge both locally stored catalogs.". Currently I have only one catalog and import all photos into one single (meanwhile huge) folder on my harddisk.
Have a great day and best regards
Not really one way, but look into File > Import from Another Catalog. I have a catalogue on the Mac Air laptop, then upon return use File > Export as Catalog to make a copy on an external hard drive (there are options to include previews and the originals or "negatives"). I then attach the hard drive to the desktop PC and File > import from Another Catalog and point to the EHD. It then imports everything done on the road.
You can also use these two commands to roundtrip - ie copy stuff from the desktop, work on it while on the laptop, and then merge it back to the desktop afterwards.
I mention the EHD only because of the file sizes. Copying over the network or Dropbox is also possible.