No, 'for the sake of' is not a formal phrase : it's a normal everyday phrase. We use it all the time in various contexts in our daily lives e.g. "My gran and grandad only stayed together for the sake of their kids". On the Cambridge Vocabulary Profile, it is B2. This means that an average B2 candidate would be expected to know and use it.

Not particularly formal. The Collins Dictionary is an excellent resource for English learners. One feature is that it shows the word frequency. [most common 1,000; 4,000; 10,000; 30,000; beyond 30,000; ] "Sake" is in the most common 4000 words and "for the sake of" is given as a dictionary example. sake https://www.collinsdictionary.com/dictionary/english/sake noun 1. benefit or interest (esp in the phrase for (someone's or one's own) sake) 2. the purpose of obtaining or achieving (esp in the phrase for the sake of (something))