What is the difference between bootstrap sampling vs multinomial distribution? I feel like bootstrap sampling and multinomial distribution sampling are equivalent. Just want to verify whether my understanding is correct.
Say my data is 1, 1, 2, 2, 2, 2, 3.
Multinomial distribution sampling would be sample 7 observations from a multinomial distribution (2/7, 4/7, 1/7). Isn't bootstrap sampling with replacement essentially the same thing because every time we draw an observation from the data, the chance of getting each value is also (2/7, 4/7, 1/7)? Thanks in advance

Yes, you can think of it as drawing from a multinomial distribution. In fact, when I code bootstrap procedures from scratch, I do exactly that over the indices of my data.
library(MASS)
set.seed(2022)
N <- 100
B <- 1000
X <- MASS::mvrnorm(N, c(0, 0), matrix(c(1, 0.9, 0.9, 1), 2, 2))
for (i in 1:B){
  
  idx <- sample(seq(1, N, 1), N, replace = T) # This is multinomial sampling
                                              # with each index "category"
                                              # having an equal probability
                                                
  X_boot <- X[idx, ] # Select the indices
    
  # Then do something with X_boot, such as calculating the correlation
}

Since you have duplicated values in your 1, 1, 2, 2, 2, 2, 3, drawing uniformly over the indices is in some sense equivalent to doing a multinomial draw with $P(1)=2/7$, $P(2) = 4/7$, and $P(3)=1/7$. There's this issue where the values 1, 2, and 3 are numbers and not categories, so it is debatable if this is multinomial, but this technicality can be resolved by doing a distribution like:
$$
P(\text{Pick 1 and add it to the bootstrap sample}) = 2/7\\
P(\text{Pick 2 and add it to the bootstrap sample}) = 4/7\\
P(\text{Pick 3 and add it to the bootstrap sample}) = 1/7\\
$$
