Not really. I mean, if Google chose to let them than sure. Otherwise it wouldn't be that hard to have Chrome (which is based on an open source project but is not itself open source) send an encrypted key in each packet ensuring it came from a Chrome browser.
That said both Microsoft and Apple have voice recognition engines built into their client OS which seems like a much better option latency wise.
If they're going to support it in Chromium, they'd have to support it for all browsers, else they'd get bad PR, as Chromium is open source.
I was speaking in the theoretical. Honestly I think Google would be overjoyed if other browsers decided to make themselves reliant on Google's server farm to function. I think that's why the issue isn't really addressed in the proposed standard...
And other browsers could just extract the key and ship it as well.
I'm assuming there would be some kind of algorithm to generate the encrypted key.
The algorithm has to be seeded with a secret (Stream ciphers work this way). You can't get around having to have Chome own a secret, and it's really hard to protect a secret when you're in a hostile environment (e.g. the user's PC). This is part of what makes DRM really hard.