Jews still owned Jewish slaves in the Talmudic era because Talmudic authorities tried to denounce the practice that Jews could sell themselves into slavery if they were poverty-stricken. In particular, the Talmud said that Jews should not sell themselves to non-Jews.
The Israelites and Jews had slaves throughout the biblical period, as various passages of the Bible attest. Exodus 12:44 talks of the purchase of slaves and the need for them to be circumcised, which implies that these were non-Hebrew slaves.
As the question implies, it seems anachronistic that the fleeing slaves would have had slaves at the start of the Exodus or even at any time during the forty years of wandering, but Exodus 12:44-45 does not necessarily mean that the fleeing Israelites actually had slaves at this stage -- it can be read as a teaching or instruction for when the Israelites would soon have slaves.