It depends. It could have discreet op-amps for each channel, or there could be one op-amp for each group of 4 channels (four amplifiers in one chip).
And all the amplifiers are going to share a common power supply, so if there was noise coming from there it would be distributed among all channels. This is a pretty common problem, and all it takes is one leaky capacitor to start making trouble.
Either way, it's likely an analog rather than a digital problem. And I wouldn't worry too much about it, as 40KHz is easy to remove without affecting sound quality. For starters, don't record at 88.1 or 96 KHz as that alone will assure no 40KHz content.