There's some similarities with the FDA, but quite a lot of important differences. NHTSA doesn't approve vehicles, for example. Manufacturers can sell whatever they want. NHTSA simply has the power to issue recalls (preventing further sales) if those vehicles don't comply with FMVSS.

NHTSA also doesn't incorporate standards like the FDA does, so while they're aware of industry standards and employ a number of relevant experts for various purposes, you're under no obligation to follow them. Tesla is actually an example here. Their development processes don't follow ISO-26262 (the automotive equivalent of IEC-62304), though stating this properly would need a lot of asterisks I don't want to get into.

The EU does both of these things for vehicles, though it's a bit more complicated than a flat approval or rejection and it's handled by a designated third party that also does medical device testing like TÜV SÜD. Other countries like the UK have a dedicated agency to handle type approvals.

Emissions related components work very similarly, replace the software and it’s presumed to be a defeat device unless proved otherwise.