Here's an EE-student's perspective, having taken classes both in continuous-time and discrete-time signal processing:
Any time you use any kind of frequency-response (more specifically amplitude response)-changing filter (analog or digital), you're going to alter the phase response (versus frequency) as well. It's not a function of imperfections in analog gear, or the lack thereof in digital signal processing, it's a function of the relationships between time-domain and frequency domain (both amplitude and phase versus frequency)
Don't confuse phase shifts with digital processing latency (even phase shifts they can and do cause, and are actually the very directly related to, time delays). They are very different things (though, as above, any time delay is invariably related to a phase shift, and vice versa)
If anyone wants to learn more about the relationship between time-domain, and frequency-domain representations of signals, do some reading up on Fourier Series/Transform, which relate time-domain to frequency-domain signal representations. Actually, knowing a little bit about fourier transforms is very useful in understanding a lot of things about audio - for example all the arguments about sample rate and Nyquist frequencies (although technically you'd want to look at the Z-transform for this discrete time discussion, but Fourier is pretty analogous)
Any time you're changing something in one domain, you're changing it in the other - changing the way a signal is in the time-domain (amplitude versus time) changes it in terms of amplitude versus frequency, and phase versus frequency
(of course some changes could presumably result in some of these things having a change of zero - i.e. change the frequency responses amplitude but not phase, but that's really not the issue here).
When we talk about linear phase filters, that doesn't mean that the phase shift if THE SAME over all frequencies, it means that the phase shift versus frequency is LINEAR - if you double the frequency, you double the phase shift, etc. If we shifted all frequencies by the same PHASE ANGLE, they'd still get all whack in their time relationships. The benefit of linear phase is that if you look at the appropriate transformation, this linear phase shift versus frequency corresponds to all frequencies being delayed in time by the same amount - the whole signal is just time-delayed. This can be shown by examining the Z-transform of a digital signal (The transform in discrete time - aka "sampled" - that corresponds to the Fourier transform in continuous time). We find that z^-1, inverse of the complex discrete-time frequency variable z, which is used in (one way of) describing a digital filter, is actually equal to a delay by one sample.
The ability to create a linear phase filter is one of the important benefits of digital filters. Another being the ability to create a filter with any number of poles or zeros by simply changing DSP software functions, rather than adding costly components.