In most modern switched or hubbed Ethernet over twisted pair systems, neither the MAU nor the AUI interfaces exist (apart, perhaps as notional entities for the purposes of thinking about layering the interface), and the category 5 (CAT5) (or better) cable connects directly into an Ethernet socket on the host or router.
However, the tradition of using a separate low-level I/O device in networking has continued in fast optical fiber network interfaces, where the GBIC, XENPAK, XFP, and enhanced small form-factor pluggable (SFP+) pluggable transceiver modules using the XAUI interface play a similar role.
So while MII and AUI as external connections were similar from a user's point of view, in terms of signalling the equivalent of MII for classic Ethernet were the non-standardized interfaces between MAC and Serial Interface as found in e.g. the Am7990 family.
The purpose of a MAU is to provide the physical means for communication between local network data link entities.
As a physical interface, an MAU can be implemented independently among different hardware manufacturers and achieve the intended level of compatibility when interconnected in a common local network.
In normal mode, the MAU functions as a direct connection between the baseband medium and the data terminal equipment (DTE), enabling message traffic between stations.