On iOS, Tweetbot was updated to use a Safari extension to allow this to happen, but the Mac version lags far behind and doesn't support this. There's also no setting in the Twitter app to change this behavior. Maybe Tapbots will one day update the Mac app, but who knows.
I have a suggestion, but I don't know if it will work. Go to https://github.com/Lord-Kamina/SwiftDefaultApps and download Swift Default Apps. It loads as a pref pane in System Preferences. Then open that pref pane and go to the Applications tab. Look at ALL the URI's that open in Twitter.app, and then make sure that they are unchecked in Twitter and you check them for Tweetbot.
Thanks for letting us know. I have the pref pane installed, but not any private Twitter apps. I suspect Twitter does this because it wants to control their app, similarl to how they restrict developers from full access to all their APIs. No idea how that might change in the Musk era, either. I do prefer the Mac App to using the browser, so I'm using it, but there used to be three independent apps that I liked better: Tweetbot, Twitterific, and Your Fukurou. And we all know Tweetie was bought and transformed into the Twitter app.
This is all on Twitter, and I suppose how we feel about this situation will be one factor in determining how much we use Twitter.