How to add home directory path to be discovered by Unix which command?

That line in your .profile should be one of

export  PATH="$PATH:$HOME/Unix/homebrew/bin"
PATH="$PATH:$HOME/Unix/homebrew/bin"
PATH=$PATH:$HOME/Unix/homebrew/bin
PATH=$PATH:~/Unix/homebrew/bin

The ~ character is only expanded to your home directory when it's the first character of a word and it's unquoted. In what you wrote, the ~ is between double quotes and therefore not expanded. Even if you wrote export "PATH=$PATH:"~/Unix/homebrew/bin, the ~ would not be expanded because it is not at the beginning of a shell word.

There is a special dispensation, which is intended to write values for PATH and similar variables. If ~ is just after the equal sign that marks an assignment, or if ~ is just after a : in the right-hand side of an assignment, then it's expanded. Only plain assignments have this dispensation, export PATH=… does not count (it's a call to the export builtin, which happens to have an argument that contains a = character).

Here, you do not need to export PATH because it's already exported. You don't need to call export when you change the value of a variable (except in old Bourne shells that you won't find on OSX or Linux). Also, in an assignment (again, export does not count), you do not need double quotes around the right-hand side, so PATH=$PATH:~/Unix/homebrew/bin is safe even if $PATH contains spaces.

This is happening because ~ has not been expanded. Your shell knows how to deal with this, but which does not (nor would most other programs). Instead, do:

export "PATH+=:$HOME/Unix/homebrew/bin"

Alternatively, stop using which, and use the (almost always superior) type -p.

Here is a demonstration of the issue:

$ echo "$PATH"
/usr/local/bin:/usr/bin:/bin
$ export "PATH+=:~/git/yturl"
$ yturl
Usage: yturl id [itag ...]
$ which yturl
$ type -p yturl
/home/chris/git/yturl/yturl
$ export "PATH=/usr/local/bin:/usr/bin:/bin:$HOME/git/yturl"
$ which yturl
/home/chris/git/yturl/yturl

Bear in mind that some other programs that look at $PATH may not understand the meaning of ~ either, and take it as part of a relative path. It's more portable to use $HOME.