The only similarity between these two UX scenarios is that they involve encrypted network protocols. From a user standpoint there's no similarities.
Firstly, the messaging decision is presented to the user before an action (send SMS/Signal). It's capable of blocking and takes place as part of an active use flow where the user is trying to complete a task. With browsers, the differentiation in UI is displayed after a user action. It doesn't block and the user doesn't require interaction to achieve any goal. Why on earth should they pay any attention to it?
Secondly, the UX for messaging is an equivalent paths binary decision: you're asking people to choose A or B. There isn't an inherent default so a user doesn't start out with a bias toward one or the other. They can easily be required to read to proceed.
With browsers it's a yes/no binary decision: the default (yes) is insecure (for an insecure website). It requires no action from the user. The secure option (no, leave) asks the user to do something. It's a choice between inaction (insecure) or action (secure). That's heavily stacked.
Lastly, even the context surrounding the apps themselves is incomparably different. One is a security upgrade of an application everyone's been using for decades (often unknowingly; "the icon for the internet"). The other is an app people consciously download and install explicitly for security reasons (regardless of whether they understand those security reasons it's at least the motivating factor).