The MFA card requires knowledge of the current time, if I understand them correctly... thus there must be a very low power clock chip inside it, along with the CPU, etc.
Battery life is specced for 3-5 years mostly.
The scale is also using some sort of internal battery. It could be a gel type battery or a CR2032 or similar inside. My CR2032 powered scale is going on 12 years with the original battery.
TL;DR: there’s batteries inside.
a scale is certainly large enough for a battery to be inside.