Trustwallet and some other wallet that I can not point for people to use also have that, but I do not think it is working in a way that it is pointing to phishing sites, I think it is pointing new sites. I have seen new created legit site for presales that would be first tagged as may not be legit but legit, but as the site grows older, the warning message will not appear again. But why should wallet become a browser? That browser called web3 is noe majorly used by scammers to make novice tí connect to their fake website, we can see people complaining everywhere and including this forum about his they connect to fake website and all their money stolen.

Metamask is not a shield in anyway. If you have your coins there, there will not be warning if a hacker have access to someone's metamask wallet and known the person sees phrase. Metamask can be as vulnerable as other online wallets.