Yes Indeed
I can confirm that the CDRW in the MRS 1608 is nothing more than a standard EIDE drive like your computer uses.
Seems to be a common trick in UK to sell the MRS 1608 + Free CDR. Basically they sell you a 1608 and an aftermarket CDRW as a pack. It's cheaper than a 1608CD.... I just bought one.
The CDRW supplied with mine was a standard Lite-On PC CDRW drive (even came with all the PC burning software and cables to hook to your soundcard etc!). Installs exactly like a PC drive. Remove the blanking plate slide in the drive, connect the power + data connectors, and secure with the 4 mounting screws. All screws, cables & full instructions are supplied with the MRS... If you can read and you can hold a screwdriver, you cannot possibly get it wrong, the 2 cables can only be inserted one way up.
I cannot imagine that there is any practical reason why you could not use any EIDE CD drive. There may be certain features/speeds the drive needs to support but I can't imagine that'd be an issue with most modern drives.
The differences between most aftermarket PC products (CD Drives included) is simply 'badge-engineering' - most 'manufacturers' are sourcing their stuff from the same place.