The two systems, namely, Gaia BH1 and Gaia BH2, that have been confirmed as dormant (i.e. no X-ray emission detected) black hole (BH) – low-mass star binaries in the latest Gaia mission data release are intriguing in the context of their formation and evolution. Both systems consist of 9M BH and 1M star orbiting each other on a wide, eccentric orbit (e 0.5). We argue that formation of such Gaia BH-like systems through the isolated binary evolution (IBE) channel, under the standard common envelope assumptions, and from dynamical interactions in young massive and open clusters are equally probable, and that the formation rate of such binaries is of the order of 10 M for both channels. We estimate that, according to our models, there are at most 900 detectable Gaia BH-like binaries in the Milky Way thin disc. What plays an important role in formation of Gaia BH-like systems via the IBE channel is the mutual position of the natal kick velocity vector and the binary angular momentum vector. We find that natal kicks with a median magnitude of 40 km s are preferred for the formation of Gaia BH1-like binaries. Approximately 94 per cent of those binaries are formed with the BH spin misaligned to the orbital axis by less than 40. Gaia BH2-like binaries form if the low-velocity natal kick (of median magnitude 20 km s) is directed within 15 about the orbital plane. In addition to natal kick, we also discuss the influence of tidal interaction and the adopted common envelope parameter prescription on the evolution of Gaia BH-like binaries. We follow the subsequent evolution of the binaries, once formed as Gaia BH1 and Gaia BH2 systems, to investigate their connection with the low-mass X-ray binary population.