I don't see why playback on the same machine that recorded the tape should have any particular advantage, assuming the machine is alligned properly. The record and playback electronics are independent of each other and the playback head on the record machine may not be as good or in as good condition as another machine. I have two ATR-102's, one which I use to record and the other I use for playback. My playback machine has superior playback electronics (Doshi 3.0 tape prepro) to my record machine which has stock Ampex playback electronics - quite good but not at the level of the Doshi. I frequently listen to the playback of my record machine, monitoring the tape that I am recording. Then, later, I play back the same tape on my playback machine. Easy to hear the difference.
If you only align the heads once in a life time, as the recording head is aligned to the playback head you can have optimal head alignment. However next time the heads are aligned you loose this confidence.
In theory you should not play the tapes in the machine you recorded it - all mechanical systems have issues at certain frequencies, characteristic of the machine. If you record and play in the same machine the problems become twice as big. If you use another machine the probability of overlaps is much reduced. BTW, I have read that we should never measure wow and flutter with a tape recorded in the machine we are checking.
No direct experience with the situation, just what I have read elsewhere.
What about playing a tape on a different make and model than the deck on which it was recorded? For example, recording on an Ampex ATR-102 and playing back on a Studer A820?
I have the same tape for the final measurements. But for long or preliminary tests connecting a 3150 Hz tone to the recording input and looking at the output in the spectra analyzer while recording is very tempting and rewarding.
What about playing a tape on a different make and model than the deck on which it was recorded? For example, recording on an Ampex ATR-102 and playing back on a Studer A820?
I'm not sure what is ideal, but my guess is that the vast majority of tapes (15ips 2 track) that have been commercially produced since the beginning of the 15ips2track tape revolution a decade ago have been made on Ampex ATR-102's - since the two big producers of these tapes - Tape Project and Acoustic Sounds - have banks of ATR-102's as their reproducing machines. On the other hand, I would guess that only a very small fraction of the machines that are playing these tapes are ATR-102s. Other companies are using a very wide range of reproducing decks, from several different versions of Studer machines to Ampex ATR-102's, Technics 1500's, Sony, Mara MCI, and others I don't remember.