Inky

Yes - it is true, in a way. The requirement is not part of the K-1 visa (or any visa for that matter), it's a requirement of the I-94 issued.
Your status is controlled by the I-94. If you go out of status, you could have issues with ICE, and you are accruing 'unlawful presence' . Go 180+ days and you are looking at a ban if you leave (even if the AOS is in process and your granted an AP).
Only the AOS stops you from getting this 'unlawful presence' and getting days counted against you. (stops the clock so to speak)
Being out of status will not prevent you from doing the AOS however - but it's best to get it done prior to the I-94 expiring (or as close as you can).
(ICE issues, medical may expire (requiring you to do the whole medical again), the off chance that the AOS is denied, and the immigrant has to return home to process, and if they have 180+ 'unlawful presence' , ban time...*)
All spelled out here.
(*disclaimer - most people will likely have no issues like I have described above, but are you willing to take the chance? It's basically a risk vs rewards question - there are posts here on this forum where all things like this has happened.)

You guys will very likely want to file what is called DCF (Direct Consular Filing) directly with the US Consulate in Frankfurt.
A brief overview can be found HERE. This lets you guys file while you're both in Germany.