Let a, b be constant, the correlation between X&Y is the same as the correlation between aX and bY. Why is it so intuitively? I understand how to derive the conclusion mathematically. However, if a is positive and b is negative, the new correlation between aX and bY intuitively will be the opposite of the correlation between X and Y. Just windering where I am wrong in the reasoning.

Your reasoning is fine. It is the claim of equal correlation that is faulty.
library(MASS)
set.seed(2022)
N <- 10
a <- 2
b <- -3
X <- MASS::mvrnorm(N, c(0, 0), matrix(c(1, 0.5, 0.5, 1), 2, 2))
x <- X[, 1]
y <- X[, 2]
cor(x, y)
cor(a*x, b*y) 

I get equal magnitudes of correlation but opposite signs, disproving the claim.
Maybe an easier example (that skips the MASS call) is to simulate independent $x$ and $y$, calculate $cor(x,y)$, and compare to $cor(x, -y)$.
set.seed(2022)
N <- 10
x <- rnorm(N)
y <- rnorm(N)
cor(x, y)
cor(x, -y)

