A web site cannot see a user's device's MAC address. In fact, a MAC address is part of the low-level network stack and is not propagated from one subnet to another.
The MAC address, by TCP/IP standards, is never communicated outside of the local-area network to which it pertains -- routers beyond that LAN don't even get the information you're trying to record. So, no.
It would be theoretically possible if phpBB were running on an internal network (intranet), but then an IP ban would be just as effective, assuming no DHCP.
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