How does Bitwarden protect against a malicious mobile app update?
Although how could any product protect against that kind of attack? If the supply chain is compromised, it's pretty much a wrap.
This is not specific to Bitwarden though, it's the same for all apps. Now because Bitwarden is open source, you can actually compile and install it yourself (if you're confident that the sources you are compiling are the legitimate ones, and not a fork that will steal your passwords ;-)).
Same applies for e.g. Signal Messenger: since it's difficult to check what version of the code is coming from your store, you can always compile it from source and install it yourself.