If the SIM or router IMEI is linked to your real identity, then you accomplish nothing by using a portable router.
If your phone's IMEI is not linked to your identity, then also the portable router adds nothing.
If the SIM and router are not tied to your identity, but the phone IMEI is, then I can see some privacy benefit to hiding the IMEI from the cellular network.
But honestly, I really don't understand why so many people are asking about portable routers. If you really need to hide your identity from the network, get a used phone or pay cash or something. Or just don't use mobile networks.
Maybe someone can enlighten me.