How to correctly return values in shell script function for edge cases?

There are a number of problems here; #1 and #3 are the ones that directly address your question.

1. When a function/command/whatever may need to print both regular output (e.g. the path to an animal directory) and error/status output (e.g. "inavalid animal"), it should send the regular output to standard output (aka stdout aka FD #1, the default), and error/status output to standard error (aka stderr aka FD #2), like this:

   echo "Invalid animal" >&2    # This is sent to stderr
   

2. Generally, functions should return rather than exiting. If a function does exit, it exits the entire shell, but in this case the function is running in a subshell due to $( ), so it only exits that. Using return avoids this inconsistency.

3. When a function/command/whatever may fail, you should check its exit status; there are a number of ways to do this:

   if result=$(getAnimalFolder); then
       : # command succeeded!
   else
       echo "OMG it failed!" >&2
       exit 1
   fi
   

   or

   result=$(getAnimalFolder)
   if [ $? -ne 0 ]; then    # $? is the status of the last command
       echo "OMG it failed!" >&2
       exit 1
   fi
   

   or

   result=$(getAnimalFolder) || {
       echo "OMG it failed!" >&2
       exit 1
   }
   

   I use the last form a lot, since there are a lot of steps in a script might fail, and having a simple & compact way to include the failure handing code makes the overall script more readable.

4. In general, functions should take their input as arguments rather than via global variables. So in the function you'd refer to $1 instead of $ANIMAL, and you'd run the function with something like:

   result=$(getAnimalFolder "$ANIMAL")
   

There are also a number of basic syntax errors and bad scripting practices in the script: don't put spaces around the equal sign in assignments; do put double-quotes around variable references; don't use all-caps variable names (to avoid conflicts with the many all-caps variables that have special meanings); do check for errors on cd commands (if they fail, the rest of the script will run in the wrong place); and when comparing a single variable against a bunch of values, use case instead of a bunch of if elseif etc.

shellcheck.net is good at recognizing many of these common mistakes. Strongly recommended.

Here's what I get with all fixes in place:

#!/bin/bash

getAnimalFolder() {
    local animalname=$1
    local animaldir=""
    case "$animalname" in
        lion ) animaldir="./animals/lion/" ;;

        tiger ) animaldir="./animals/tiger/" ;;

        cheetah ) animaldir="./animals/cheetah/" ;;

        * )
            echo "Invalid animal: $animalname" >&2
            return 1 ;;
    esac
    echo "$animaldir"
}

read -p "Give me an animal: " animalname
result=$(getAnimalFolder "$animalname") || {
    exit 1 # Appropriate error message has already been printed
}
cd "../$result/age/" || {
    echo "Error changing directory to ../$result/age/ -- aborting" >&2
    exit 1
}