I'm not a current Cambridge student but I did study MML from 2015 to 2019 – my honest answer would be that you do need to really enjoy your subject and be prepared to work at it to be able to succeed at Cambridge. Some of the best advice I was given was that you basically have a similar workload as you would for a full-time job. If you genuinely enjoy your subject then that workload is manageable and enjoyable, but it isn't necessarily for everyone.
You do absolutely have time to socialise and do societies, sports and the like – just as people who have full-time jobs still do those things. And whether it's 'normal' or not, you're surrounded by people who are all in the same boat, so it all feels normal in the moment.