To some extent this is true. In particular, a lot of the PEQs use the same formulas for generating filter coefficients. They're the formulas from Robert Bristow-Johnson, if you google "RBJ cookbook filters" or something like that, you'll find them.
There are some PEQs that use different digital filter prototypes, but I don't remember them offhand. If you see an EQ with options for things like Chebyshev, Elliptical, or Butterworth, that is a different filter from the RBJ ones that you hear used everywhere.
Also there are some different EQs besides those, but the general statement holds. There are Digital EQs other than linear phase, i.e. there's also minimum phase.
Part of the deal is this. "Digital EQ" means a FIR filter, which is what you guys call an impulse response. It's a finite amount of N samples, that are combined with the past N input samples to form an output sample, using a mathematical operation called convolution. You can get whatever phase response and frequency response you want, but the resolution is limited by the number of samples, and more samples means more resolution, and it's computationally intensive, and much more so by using a longer impulse response. When you see a linear phase EQ, it's almost certainly a FIR filter because you can design the phase response to be linear very easily.
The other main kind of filter is IIR, which is an infinite impulse response. Here, the output sample is a function of just a few (like 3 or so) of the past input samples, but also the past output samples. It's a recursive filter, unlike FIR. IIR filters are a lot harder to design; the general technique is to design an analog filter that does what you want, and use a mathematical transformation to get the digital filter coefficients. That's what RBJ did. These are way less computationally intensive, because each sample requires like 6 multiples instead of 2000 like with a FIR filter. The downside to this is, phase response is trickier.
First of all, with an FIR filter, you can independently design amplitude and phase response. It's really simple. However, with an IIR filter, it's not straightforward like that. You can't just construct a phase response, but you can choose an analog filter with a phase response close to what you want, but there aren't as many options. Furthermore, say you have an analog filter you really like, say a nice Neve EQ circuit or something. The bilinear transform, which is used to get the IIR filter from the analog prototype that I mentioned above, doesn't totally preserve phase response! As the frequency increases, the digital filter's phase response becomes less like the analog phase response. It gets warped. So the bilinear transform is a good technique for digitizing an analog filter, but it won't give you exactly the same phase response. I'm skeptical that you can hear the warping especially cause we're less sensitive to phase differences at high frequencies.
Now, a really good model of an analog EQ unit could be given with a FIR filter, but like I said that's computationally intensive. That's why you see these IIR filters so often. And with all of these IIR filters, you're usually hearing RBJ's filter coefficient prototypes, because they're widely available. Like I said, there are Butterworth, Chebyshev, Elliptical, and some others. They all have different optimality (maximal flatness in the passband, in the stopband, etc). The reason you don't see them used as often is they're more complex to design -- it's not as simple as just plugging it into a formula like RBJ's. They are used in some places though.
I know this got technical, if you're curious just wikipedia "FIR Filter" or "IIR Filter" or "Digital Biquad" or "Butterworth filter", etc.
Edit: My bad, I confused myself. To correct my error: using the bilinear transform to make an IIR filter from an analog prototype results in a filter where the amplitude response differs slightly close to the Nyquist frequency (as the frequency gets high, error gets worse), but this difference is pretty small and can be compensated for by pre-warping and oversampling. On the other hand, the phase response, I believe, is substantially different from the analog filter.
