ATMEGA328P SPI MMC/SDIO
As you point out, there are SDIO cards available with various useful functionality.  This should tell you that it is possible to produce an SDIO slave device.  If you can find an SDIO slave interface IC, or a microcontroller with an SDIO interface that can be configured as a slave, or if you can write an HDL well enough to build an SDIO slave interface in an FPGA, then you have a shot at doing this.

More broadly, I suggest that you explain what it is you're actually trying to do.  Doing so may elicit an approach that is practical and that you haven't thought of.