Quick answer
Yes. Madeira's tap water is safe to drink across the entire island. It comes from levada catchments in the mountains, is treated to EU standards and tastes fine. Many locals prefer bottled water for habit, not safety. Refill bottles freely.
Context
Drinking water in Madeira is sourced from upland springs feeding the levada system, then treated and distributed by ARM. It meets EU Directive 98/83/EC limits and is regularly tested.
Mineral content varies by parish, Funchal's water is soft, north-coast water can be slightly harder. None of this affects safety.
Porto Santo is different: all of its public drinking water comes from seawater desalination. It is just as safe, but some visitors notice a different taste.
Plastic bottle culture is heavy on the island; bring a refillable bottle and you'll save €5–10 a day and a lot of waste.