Are iOS in-app purchases information not shared on M1 MacBook?

Each app developer decides how they validate receipts and licenses. Microsoft Office 365 designed their licenses so macOS, iOS and iPadOS all share the same one license. Apple provides shared app storage so even if the developer doesn’t want to host their own license servers and validation logic, an app developer could easily design shared licensing model via Apple App Stores.

I don’t see Rosetta Stone saying clearly in support either way which they chose, but there’s no technical reason why they don’t honor one license on all versions of the app. They could easily expect you to pay as well and Apple doesn’t really weigh in which model a developer chooses.

I would open a support ticket with Apple (since you paid them presumably for the subscription) and with Rosetta Stone (since they publish both apps) to ask how it is intended to work. Once you know their intent you can then pursue support if your license isn’t working or know they just don’t intend this to work.