Inheriting path modifications vs. Inheriting aliases

My login shell is zsh. I have added the following two lines to my .zprofile file:

path+=$HOME/foobar123
alias foo='echo bar'

Then, if I switch to bash or fish, by typing bash or fish and pressing Return, echo $PATH will continue to show me the foobar123 part, but the foo alias won't work. Why is that? Why aliases and path modifications are treated differently?