Is the word “exercise,” being a noun, countable or uncountable?
Some people say it is an uncountable noun, but why does
exercise add “s” in the phrase " do exercises"?

Depending on the context, the noun ‘exercise’ can be both – countable as well as uncountable. For example, when you refer to grammar exercises, it’s countable. When you talk about physical activity, it is uncountable as in ‘You need more exercise’.[YSaerTTEW443543]

TOEIC listening, photographs: Gardening with the dog[YSaerTTEW443543]

When used as a noun referring to physical activity:
‘exercise’ the act of exercising (uncountable)
‘exercise’ (countable) - refers to a specific movement designed to develop one aspect.
‘exercises’ (countable) - refers to a set; a series of specific movements designed to complement each other in providing a workout for either the whole body or for a specific area of the body.