Can't install client certificate on different device?

Firefox does not support importing private keys from "PEM" files.

When you ask it to import a certificate from a PEM or DER file, it'll import only the certificate, not any private keys that might be in the file. Instead, it expects to associate the certificate with an "unbound" private key that it already has (i.e. one that was previously generated using the obsolete <keygen> tag).

If you have a certificate with an externally generated private key, you need to export both to a .p12 file in the PKCS#12 format (aka PFX, just like the .pfx files used by Windows), e.g. using openssl pkcs12 -export.