Come stay
a while.
What this is
Fog & Company is a private booking page for one person's friends and family. Guests sign in, ask to be let in, and once the host approves them they can see which nights are free, request a stay of their own, bring up to two other people and leave a note. The host accepts or declines each request, and keeps dates for themselves when they need the room.
Why it asks for a Google account. Signing in with Google tells the site who you are: your name and email address, so the host knows whose booking is whose. That is the only thing it is used for.
Why it asks for Google Calendar. Only the host is asked for this, and only to write confirmed stays onto a calendar of their choosing and send an invitation to everyone on the booking. Cancelling a stay deletes that event again. The app never reads or changes anything else on the calendar, and guests are never asked for calendar access.
Nothing is sold, advertised against, or shared beyond the services needed to run the site. The full detail is in the privacy and the house rules.