Which Arduino-compatible microcontroller for 2 continuous analog audio recording and 2 digital pin snapshots every second?
My question therefore is would a ATmega328P or ATmega2560 be suitable for such requirements


No. Not in the slightest.

Your primary requirement of sampling audio at, basically, CD quality (or better) requires the use of a proper CODEC chip - and to drive that you require a microcontroller that has I2S. None of the 8-bit AVR chips have that.

On top of that you will need a LOT of memory. Far far more than an 8 bit Arduino has.  Two channels of 16 bit audio at 44100Hz (CD quality) requires 176400 bytes per second. 

You will need double that.

To both continuously record and save the data you will require DMA to do the recording.  The DMA reads the incoming data from I2S and stores it in one buffer. When that buffer is full you generate an interrupt and switch to a second buffer. You can then save the first buffer to SD - assuming that it can be done in under a second.  Note that SD cards, when controlled by SPI as they generally are on microcontrollers, are generally quite slow.

So with all that in mind:


  I do like the idea of working on a AT91SAM3X8E chipset, but I'm afraid this might be an overkill for such a small project


Overkill? No. Not suitable since it is not up to the task. Yes, it has I2S. Yes, it has DMA. But it doesn't have anywhere near enough memory.  You need for the audio alone 350 kilobytes (the SAM3X8E has up to 100 kilobytes) - not to mention memory required by the SD card's buffers.

So you're looking for a microcontroller that has full I2S support, DMA, and around half a megabyte of memory.  And those aren't small and cheap like the Arduino's 8-bit AVRs.