Safe, yes, but desalinated
Malta has almost no fresh water of its own. More than half the tap supply comes from reverse-osmosis desalination plants, the rest from heavily mineralised groundwater. The result is water that is perfectly safe to drink and constantly monitored, but with a pronounced salty, chlorinated taste. It is not a cleanliness question: it is the taste that puts people off.
What the Maltese do
They buy bottled water, or fit a filter at home. That is the general habit on the island, families included. In restaurants you will be served bottled water by default. In hotels, asking for a jug of tap water is an unusual request. Nobody will look at you oddly for drinking it, but expect an unfamiliar taste.
What it costs
A litre and a half costs under a euro in a supermarket, and two to three euros in a bar or a beach kiosk. Over a week with several people the gap is real: buying large bottles at the supermarket rather than small ones on the beach divides the spend by three. Plenty of hotels and hostels now put filtered fountains out, which lets you refill a bottle.
The habit that matters at night
This is the genuinely useful part. A night in Paceville means five hours on your feet in a club at 30 degrees, after a day on the beach. Dehydration is the main cause of ruined mornings in Malta, well ahead of the alcohol itself. Clubs serve water free or for a couple of euros at the bar: a glass of water between drinks changes the end of the night and the next morning completely. In summer, a bottle in the bag before you go out is basic.
Brushing teeth, cooking, showering
No problem at all. Tap water is fine for brushing your teeth, showering, cooking and boiling pasta. Its high limescale content furs up kettles faster than elsewhere and can dry out skin and hair a little, but it presents no risk.