Is the vacuum pump and brake booster the same thing

@summerwiles I can tell you that they are not the same things. Brake booster and vacuum pump are two separate things. The vacuum pump removes air from the brake booster. The particular model 2013 BMW 328i had issues with the vacuum pump on their new models and had reports of brake failures.

Looks like the part number for the vacuum pump is 11667622380 and looks like this Example only not an endorsement

BMW calls the brake booster the brake servo unit. The part number should be 34336851099 unless your vehicle has M sport brakes, then the part number is 34336851098

Parts are widely available from many sellers, even places like Autozone etc. Always verify the numbers and fit with the seller.