You're asking us to speculate about something that hasn't been released yet.
However, looking at the FAQ, it's clear that this is just a fancy Linux box with some emulators on it.
Processor: Intel Coffee Lake S Series Processor
Memory: 2GB DDR4 RAM
Polymega’s Modules use top tier emulators with low latency controller inputs.
Emulators: Legally licensed versions of Mednafen, Mesen, Kega Fusion, and MAME with additional bug fixes, CD BIOS development, and replaced YM2610 for Neo Geo CD from Playmaji.
From their marketing text and FAQ, it's clear they want you to copy your cartridges into the internal storage (or buy them from whatever partner they've rustled up) and play them from there.
It's a faster Ouya, in other words.
So how would it 'support' existing cartridges?
To play existing cartridges in an emulator, they first have to be dumped into RAM. To do this, the makers have to devise an interface that's compatible with the cartridges only to the extent that the contents of their ROM can be read. For Genesis games, this can be as simple as reading the mask ROM in one linear block. For Game Boy and Master System games, this involves negotiation with the mapper chip. This stage can be implemented relatively easily for most consoles.
To the end user, they put in a cart and then play the game. It's transparent to them: it feels like a console, and so they'd be satisfied. The cartridge slot is almost certainly not used at all during regular gameplay on this system.*
The important distinction is that the reading interface doesn't have to be particularly fast. To run a game directly from a cartridge requires accessing the data and acting upon it in real-time. To dump a cartridge can be done leisurely.
* Reading and writing the save RAM on a cart could be somewhat straightforward, though would need to be tested on a game-by-game basis since it's not homogeneous (at least on Genesis). Impressive if they bother. :)
How does being an emulator 'stop' Game Genies and Everdrives working?
Everdrives, Game Raccoons, etc. interface with external hardware like (micro/)SD cards themselves, and have quirks beyond the standard Nintendo/Sega interface that aren't emulated in most emulators - Flash memory bankswitching, etc. If makers of this Linux console haven't programmed support in for these, it doesn't exist. It doesn't come for free. And support in this case would be some kind of bidirectional software passthrough from the emulated Everdrive menu to the real cart...? Why bother?
If this was a hardware-based console, using FPGA or real chips, it could reproduce the exact bus interface and timing, you'd have compatibility with save RAM, enhancement chips, Game Boy Camera, real-time clocks and the like through that interface. The situation is very similar to "How does the Everdrive handle all the special chips and stuff that were put in cartridges?", except reversed: the ersatz console would have to correctly talk to all the real enhancement chips.
There'd be little reason for the makers of that Linux emulator-based console to support any of this, given their stated monetising approach.
It's much more likely they'll support a simulated Game Genie on their emulated console, since cheat support will come as standard in the emulator set they've taken, but that's all.