Not sure I understand completely but will attempt to answer:
There would be varying situations in which you can use Machform, two of which are presetned below as the typical uses (for self-Hosted):
1) You have installed Machform on one site (domain), and then can embed the forms in any site you wish (and as many as you wish). That is using the single-site license. Set up of Recaptcha then requires one set of the Site and Private keys and the Recaptcha will work on any form you embed on any site, where they run from that instance of Machform.
2) You install Machform, separate installations of the MF application on each site and the forms are only embedded in that site (e.g. www.domain2.com, www.domain3.com etc). In that case you'd have separate instances of Machform running on each of the websites (domains) and requires the multiple-site license. Recaptcha on each one would require separate Site and Private keys. You only have/need to enter one set of keys, once in each Machform installation.
I hope that's clarified and answered your question.
Update: you might need separate sets of the Recaptcha keys, or i think there's even a 'global' set of keyst that would work on any site, or indeed any PHP application that uses Recaptcha V2.