Is there anything stopping web browsers from snooping into our activities? I don't mean web history or any of that generic stuff (Chrome already collects them, right?), but is it technically possible for a web browser to intercept what is being entered in a secure page (or let's say the password field)?
Is there anything stopping from web browsers snooping into our activities?
Nope, not really. That's a bit like asking
Is there anything stopping my keyboard from recording my keystrokes?
In theory, we are interacting directly with the keyboard / web browser, which in turn talks to other components for us, and we are trusting that it is acting in our best interest.
You could do as @StackAbstraction suggests and monitor the network traffic in and out of your browser looking for suspicious exfiltration, but you're going to lose this game since there are many many sneaky ways to exfiltrate data.
Fortunately, both Firefox and Chromium ("the open-source web browser project from which Google Chrome draws its source code." -- Wikipedia) are open source. With some google-diving, you can probably find 3rd party security audits of the source code.
P.S. If you think I'm being dramatic to draw a parallel with trusting your keyboard, check out the Razor "Stats & Heatmaps" feature that they quietly snuck into their Synapse 2.0 keyboard driver in 2012. It logs all your keystrokes and uploads them to a Razor server so you can "analyze your gameplay". They removed it due to public outcry, but according to this thread they've put it back at some point.
For most major browsers that are used in the tech community, the only thing stopping them from being evil is the fact that they rely on open source code, and are widely poked and prodded.
Unfortunately, these browsers also include add-ons and extensions, which are less likely to be well-vetted and so even more likely to be causing insecurity.
It is indeed
technically possible for a browser to know what you enter in a password field.