No, although both have the same purpose: to reduce the visibility of your Social Security number on government documents.
The new Medicare number is a new number that doesn't include your Social Security number. New cards with the new numbers are being mailed out now.
The BNC (Beneficiary Notice Code) is unique to every document. Here's the government's explanation of the BNC:
In concert with the Centers for Medicare and Medicaid Services’ efforts to remove the SSN from Medicare cards, in FY 2018 we plan to replace SSNs with BNCs on benefit verification notices, which account for approximately 11 million mailed documents. We will also replace SSNs with BNCs on approximately 28 million Social Security post-entitlement notices, and on certain mailed documents sent to appointed representatives, which account for an additional 2.6 million mailed documents.