Can a shell script set environment variables of the calling shell?

Use the "dot space script" calling syntax. For example, here's how to do it using the full path to a script:

. /path/to/set_env_vars.sh

And here's how to do it if you're in the same directory as the script:

. set_env_vars.sh

These execute the script under the current shell instead of loading another one (which is what would happen if you did ./set_env_vars.sh). Because it runs in the same shell, the environmental variables you set will be available when it exits.

This is the same thing as calling source set_env_vars.sh, but it's shorter to type and might work in some places where source doesn't.

Your shell process has a copy of the parent's environment and no access to the parent process's environment whatsoever. When your shell process terminates any changes you've made to its environment are lost. Sourcing a script file is the most commonly used method for configuring a shell environment, you may just want to bite the bullet and maintain one for each of the two flavors of shell.