Is it normal to drink shower water??

I think the primary concern is that things dissolve more quickly in hot water than in cold, so if there's any contaminants in your pipes, you'll end up drinking more of it using the hot water.

I prefer the flavor of used bath water.

I don't do it often but when I am hungover and the gatorades empty in there , I will drink shower water . I usually use it to wash down and advil . I prefer to pee in the shower than drink the water though since my fridge gives out tasty water .

Because of the water heating system, hot water tends to contain more heavy metals. That is why people typically don't cook with hot water and why you heat a babies formula or tea, rather than mix it with hot water from the tap.

It's not weird but if I were you I'd be checking if your bathroom plumbing contains any lead in the pipes

If you drink the tap water in your sinks, then the shower water is exactly the same. If you only drink bottled water, then you probably shouldn't drink the shower's water. I assume you mean the water from the shower head, not the drain, right?