Is using the font 'Roboto' mandatory in Google's material design – or can I use different fonts in a material design based website?
Google is using 'Material Design' to give their vast number of different services a unified look and feel – so for google designers using Roboto is mandatory because it's part of their design by definition.
IMHO if you want to build a material design 'inspired' website, you are of course free to quote any design patterns as you like – so using Roboto could be an option. But I would highly recommend to be rather careful of how direct you want to quote google's design patterns – you probably don't want your website to look like a (cheap) copy of a google service.
Not mandatory. But highly recommended.
Roboto was introduced and standardized by Google for supporting devices of different dpi to have readability and legibility of fonts.
You can see that many apps have their own custom fonts which are either unique to their brand or to match their site. So Google cannot make it mandatory.
It's highly recommended because in case you go for another font, you have to make sure it has good readability in case of devices of different sizes.
source: Google android documentation