Both are fine.

The correct sentence is:
"Everyone has the same number of points." Why? * "Amount" is used for uncountable nouns (e.g., water, money, sugar). * "Number" is used for countable nouns (e.g., points, apples, chairs). Since "points" are countable, "number" is the appropriate choice.

The only difference is that in #1, "points" are imagined to be a non-countable noun like "water", "hope", or "salt": "Everyone has the same amount of water" "Everyone has the same amount of hope" "Everyone has the same amount of salt". However, the partition of nouns into two categories, countable and non-countable, is almost never absolute. There are nouns that land more frequently in one category or the other, but no law commands that they must remain there. "Point" is usually a countable noun, so #2 would be spoken more commonly than #1. That does not mean that #1 is wrong. You can say #1 if you choose to think of "points" as something that can flow like water. I personally would say #2. To me, points don't feel like something that flows.

They are equally correct and completely synonymous in this case.