What translation is the OP using?
Chapter 8. Fasting and Prayer (the Lord's Prayer). But let not your fasts be with the hypocrites, for they fast on the second and fifth day of the week. Rather, fast on the fourth day and the Preparation (Friday).
IOW, the instruction was not about a fast like the hypocrites, but rather do not fast with the hypocrites on Monday and Thursday. Rather fast on Wednesday and Friday.
There was only one fast day (Atonement) in the OT, but the hypocrites were presumably "pleasing the flesh, while ignoring the spiritual". Don't fast with them, but it's okay (apparently per Didache) to fast like them twice a week.
So, "with them" was wrong, but "like them" was okay on two different days.
PS To answer the OP question, it is the Jews. The Didache is saying don't fast with the Jews. (This, if a Christian did so, amongst other things, was caustically called Judaizing.)
Second Temple Period
During the Second Temple period, daily or biweekly fastings were practiced for reasons of asceticism , especially among women (Judith 8:6; Luke 2:37; TJ, Ḥag 2:2, 77d), but also among men (Luke 18:12; Mark 2:18), or in preparation for an apocalyptic revelation (Dan. 10:3, 12; ii Bar. 12:5; 20:5–21:1; 43:3; iv Ezra 5:13–20; 6:35; Sanh. 65b; TJ, Kil. 9:4, 32b). The Jewish literature of the Second Temple period also advocates fasting as a way of atonement for sins committed either unintentionally (Ps. of Sol. 3:9) or even deliberately (Test. Patr., Sim. 3:4), or to prevent them (ibid., Joseph 3:4; 4:8; 10:1–2). These reasons for fasting were strengthened by the destruction of the Second Temple and even more by the repression of the Bar Kokhba revolt and the subsequent religious persecutions.
This passage [Mt. 6:16] suggests that Jesus, like the Pharisees and the disciples of John, believes in fasting for piety, but that such behavior should be done secretly and without fanfare. Reading these texts together suggests that Jesus’ gripe with the disciplines of John was that they made their fasting known and public, and Jesus saw this as hypocrisy. Despite this, fasting gained prominence in Christian circles and emerged as an act that reflected personal piety. In addition, many ascetic Christian communities arose that encouraged its members to avoid sex, food, and other physical indulgences that distract from leading a purely spiritual life.
So, if you're going to fast, do it privately on whatever days you decide.