How do I find the probability mass function of an individual observation for a multinomial logit model? I know what the probability mass function of a multinomial logit model is. However, I do not know what the probability mass function of an individual observation i, $f(y_i|x_i)$, is for a multinomial logit model. 
 A: The model makes probability predictions. Those are the parameters of the conditional multinomial distribution.
EXAMPLE IN R
In this example, the multinomial $y$ (five classes, equal probability of each) is unrelated to the predictor $x$. Consequently, we expect the regression output to give values of about $0.2$ for each class, since each conditional multinomial distribution is that same multinomial distribution with equal probability of each of five classes.
library(nnet)
set.seed(2022)
N <- 10000
x <- rnorm(N)
y <- rmultinom(N, 1, rep(0.2, 5))
apply(y, 1, sum)
MNL <- nnet::multinom(t(y) ~ x)
round(predict(MNL, data.frame(x = 0), type="probs"), 3)

Indeed, we get predicted probability values that are close to $0.2$.
0.197
0.205
0.201
0.201
0.196

The type = "probs" is how we get R to give us probability predictions. If we omit that, then our "prediction" is the class with the highest probability, which is not a parameter of the conditional multinomial distribution.
In the language of the equation given in the OP, let $i=1$. Then $x_1 = 0$. Also $j=1,\cdots,5$, since there are five categories. Then:
$$
p_{1,1}=0.197\\
p_{1,2}=0.205\\
p_{1,3}=0.201\\
p_{1,4}=0.201\\
p_{1,5}=0.196
$$
If we make a prediction for some $x_2 = 1$ via round(predict(MNL, data.frame(x = 1), type="probs"), 3), then we would have:
$$
p_{2,1}=0.200\\
p_{2,2}=0.205\\
p_{2,3}=0.203\\
p_{2,4}=0.197\\
p_{2,5}=0.196
$$
