It depends on the company which is in possession of the masters, the company which is licensing the rights to those tapes, the mastering engineer, etc. Some record companies won’t let the masters out of their storage facility, providing the reissue company only with a copy, either digital or analog.
By "original master tape" are they referring to the 3, 4, 8, or 16-track multitrack tape (1/4", 1/2", 1", or 2" wide), or to the original 2-track (1/4" or 1/2" wide in most cases) mixdown tape? Both are considered masters. Most LP pressings are made from a lacquer cut from not a master tape, but from a "production" or "work" master, which is a copy of the real master.
Some reissue record companies are well known to use a purely analog signal path (MoFi, Analogue Productions, Speakers Corner, Intervention, etc.), so are a safe buy.
The Beatles 2014 mono LP boxset is pure analog (the stereo boxset is not), the source the multi-track masters, which allowed the new mono mixes to be made (efforts were taken to make the new mix match that of original 1960’s mono mixes, but without the low frequency rolloff which eviscerated the sound of Ringo’s kick drum and Paul’s bass).