Rust initially started as a hobby project of a person who happened to be a Mozilla employee and later got sponsored by the foundation however it was not a language that was specifically designed with browsers in mind.
a lot of early rust design was driven by Servo - an internal mozilla project, and firefox component prototypes
How could browsers not be on his mind when his job was to contribute to Firefox as a dev?
The language's largest project before it hit 1.0 was Servo. The language wasn't designed for browsers, but it certainly was informed by them and their struggles with maintaining and developing Firefox.