You have some salty HR unfortunately. Looks like Chinaman company indeed.

As some forumer here have said, there might be an underlying issue. It could be your company is looking for a reason to let go some staff. Or it could be you and your colleague is not performing.

To answer your question can company fire you for being late? Yes they can but not so easy. You will get few warning letters, then showcause letters then they can fire you. My advice start looking job somewhere else or stay there and follow the Chinaman rules. If you good in office politic go try be friend with HR and boss, you might get special treatment.