Do small preamp tubes 12ax7 need to be matched?

It might be that in the case Chakster cites, the tube is used as a cathode follower. Or a phase splitter.

Some circuit slots are going to be more tolerant of ax7 / at7 / au7 / ay7 type swaps than others. Just depends on the circuit. Cathode followers and mu followers tend to be pretty tolerant, though 12ax7 probably aren’t the best choice for driving a line of cables & SS amps.

The Herron phono supports at7 / ax7 swaps in 2 slots to change gain +/- 5dB or so. The Rogue phono is fine with at7 or au7 in the mu follower slots (similar gain difference). If you change the type in the RIAA / MM ax7 slots then you’re going to get something that sounds very much not as intended, and not accurate, though 5751 are typically a "close enough" sub for ax7.

I once accidentally put 5814 / au7 in the V1 / ax7 / phase inverter slots of tube monoblocks (Rogue Apollo) and it actually sounded pretty good, but certainly very different and colored before I noticed the reason. It was a pleasing type of coloration, actually.

If a type sub changes the sound a lot, that’s usually a coloration because the circuit isn’t designed for that tube type. One exception to this would be using a 12bh7 in place of the stock 12au7, which usually sounds a lot better (amp must be able to provide the extra heater current), because 12au7 sucks and the 12bh7 does what it does, just better.

12AU7s are to be avoided.  In my experience, any other equivalent tube (12BH7, 6CG7, 6FQ7, even 6SN7 if you can change the socket) will sound superior to a 12AU7, in a gain topology.  This does not apply to cathode follower or phase splitter use (in an amplifier; preamplifiers don't generally incorporate a phase splitter).  In those latter cases, the 12AU7 is "OK".
I am aware that the Herron allows optional use of 12AX7/12AT7, so that the gain can be modulated thereby.  I have wondered how Herron gets away with that without a noticeable difference in SQ, at least according to its many admirers. Are there switches that must be re-positioned, for one tube choice vs the other? 12AT7s can be operated on relatively low current, so that parameter can be chosen to be compatible with a 12AX7.  But I have always felt they sound better with a tad more current than can be tolerated by a 12AX7.  Just one man's opinion based on some experimentation.
Mulveling, By "phase inverter", do you refer to what is also called a phase splitter?  In that case, I can easily believe that the 12AU7 could sub for the 12AX7.  Although the operation of the 12AU7 might be improved by increasing plate current.

I care mostly about noise and microphonics. Channel variations are what a balance control is for. Phono stages and preamps in particular deserve the quietest tubes you can find. I would use only the tube type specified by the manufacturer. Euphonic distortion is just that, distortion.