When you do export PASSPHRASE=$(cat /home/x/secret) in the shell, it interpretes the $() expression, executes the command within and puts the output of it inside of the variable PASSPHRASE.
When you place $() in the json file, however, it is read by jq and treated as a normal string, which is the equivalent of doing export PASSPHRASE=\$(cat /home/x/secret) (notice the slash, which causes the dollar sign to be escaped and treated as a literal character, instead of creating a new shell). If you do that instead and try to echo the contents of the variable it will have similar results as running your script.
If you want to force bash to interpret the string as a command you could use sh -c <command> instead, for example like this:
test.json:
{
    "PASSPHRASE": "cat /home/x/secret"
}
test.sh:
IFS=$'\n'
for s in $(echo $values | jq -r "to_entries|map(\"\(.value|tostring)\")|.[]" $1); do
    echo $(sh -c $s)
done
unset IFS
This prints out the contents of /home/x/secret. It does not solve your problem directly but should give you an idea of how you could change your original code to achieve what you need.