Proves that a part actually is what the drawing says it is, and stops the ones that are not.

A tolerance on a drawing means nothing until somebody measures the real part against it. Quality engineers run that measurement, on a coordinate measuring machine or with simpler gauges, investigate nonconformances when a part fails to meet spec, and build the inspection plans and supplier audits that catch a drifting process before it ships a batch of bad parts. It is the discipline that decides whether "built to the drawing" is actually true.
Mechanical, industrial or manufacturing engineering, with an interest in inspection and process control.
A hands-on route through inspection and metrology, without a four-year degree first.
Industrial Engineers · US Bureau of Labor Statistics, Occupational Outlook Handbook, May 2024 wages and 2024–34 projections.
The Handbook names "validation engineers," who confirm equipment and systems meet safety and quality standards, as one of the specialisms this occupation covers. It is the closest published match, but it is bundled with every other industrial engineering specialism and is not a wage figure for quality work alone.
You are the person who tells production a part is not good enough, on the day production most wants it to be. Root-causing a nonconformance properly takes longer than anyone waiting for the answer wants it to, and the pressure to sign off something borderline is real and constant. Getting it wrong in either direction, being too strict or too lenient, has consequences that show up much later.