I can't say about the Sony's but several brands can remember to use a certain mode when receiving a certain kind of signal.
For example, on my Denon, if I play a 2 channel analog signal on my CD player, I can set that mode to Pure Direct. Then, if I send a digital PCM signal thru the digital CD input, I can set the Denon to use DPL II Music mode. Or, if I play a DTS CD (yes, there are a few of those out there), the unit can be set to play it back as 5.1 DTS. And as I play my music collection thru the Denon, it plays each kind of signal back the same way it was played before.
One thing I can't do is tell it to recognize mono or stereo and do something different for each. The memory is stored as signal delivery type (analog, PCM, Dolby Digital, DTS, etc.) not as detailed as mono or stereo.
BTW, if you like surround, decoding a mono signal as DPL II will give you sound from the center speaker only. Dolby has said they could put in some kind of mono enhance mode but I never seen one. Still, most receivers have a mono mode on them for these kinds of discs/programs.
To use DPL or Neo 6 decoding on my Sony I put it in "Normal Surround" mode. Then there is a setting where you tell it what kind of surround mode to use -DPLII movie or music or one of the Neo 6 modes.
Unfortunatly this setting is applied any time you chose Normal Surround, no matter what the source. So you can't have it (for example) use DPLII movie for the TV/Sat, and then use NEO 6 music for the CD input.
Note that if it is really a mono DVD then all the sound is going to come out of your center channel speaker, at least in DPLII movie. These decoding schemes work in part by looking at the difference between the L & R signals so they really don't work well with mono sources.