Hi Magnus,
Board compatibility can be found here:
https://www.tiny-circuits.com/index.php/tiny-duino/tinyshield-compat/ Most are compatible with each other. However, some require the same pins where issues begin.
From the looks of the pictures, it looks like the Tiny Arcade is using new hardware. I imagine it will be very similar to code, but it looks like the USB, TinyDuino, and OLED display boards are now fused into one board. I think this may be the same board that was on the previous Kickstarter for the children's smart watch. The joystick board layout is different with a ribbon cable and one less joystick. I cannot see if they did anything different for SD or audio connection.
If you are new to TinyDuino, I would still go ahead and play around with the existing boards while you wait for the Kickstarter to progress. You can make a great little handheld with the current boards. Codebender has game samples to look at and use.