No. It started as a Graydon Hoare's part-time side project in 2006 and remained so for over 3 years. Mozilla got involved in 2009 once the language was mature enough to run some basic tests and demonstrate the idea. Though it is sponsored by Mozilla, Rust is developed by a diverse community of enthusiasts.
Mozilla intends to use Rust as a platform for prototyping experimental browser architectures. Specifically, the hope is to develop a browser that is more amenable to parallelization than existing ones, while also being less prone to common C++ coding errors that result in security exploits. The name of that project is Servo.
Partly due to preference of the original developer (Graydon).
Partly due to the fact that languages tend to have a wider audience and more diverse set of possible embeddings and end-uses than focused, coherent products such as web browsers. We'd like to appeal to as many of those potential contributors as possible.