The same SSH key should be able to be used from multiple clients. I have different SSH keys for different networks and they’re actually stored on an encrypted USB drive that I use from several different computers without a problem.
SSH is very picky about file permissions so I would first check all the permissions from /home/{user} all the way down to the id_rsa file itself.
SSH does not really care for group or world write permissions so make sure you chmod go-w your home directory and the ~/.ssh directory for starters. I’d also make sure they’re owned by your user chown ${USER}:${USER}.
For the SSH key itself I chmod 600 them…
If you want I’ve have additional info on how I manage my SSH keys in my answer to another SSH question.