Fast X-ray transients (FXTs) are short-lived extragalactic X-ray sources. Recent progress through multiwavelength follow-up of Einstein Probe-discovered FXTs has shown that several are related to collapsars, which can also produce $\gamma$-ray bursts (GRBs). In this paper, we investigate the nature of the FXT EP250207b. The Very Large Telescope/Multi Unit Spectroscopic Explorer spectra of a nearby (15.9 kpc in projection) lenticular galaxy reveal no signs of recent star formation. If this galaxy is indeed the host, EP250207b lies at a redshift $z=0.082$, implying a peak observed absolute magnitude for the optical counterpart of ${M_{\mathrm{ r}^\prime }=-14.5}$. At the time when supernovae (SNe) would peak, it is substantially fainter than all SN types. These results are inconsistent with a collapsar origin for EP250207b. The properties favour a binary compact object merger-driven origin. The X-ray, optical, and radio observations are compared with predictions of several types of extragalactic transients, including afterglow and kilonova models. The data can be fitted with a slightly off-axis viewing angle afterglow. However, the late-time (${\sim} 30$ d) optical/near-infrared counterpart is too bright for the afterglow and also for conventional kilonova models. This could be remedied if that late emission is due to a globular cluster or the core of a (tidally disrupted) dwarf galaxy. If confirmed, this would be the first case where the multiwavelength properties of an FXT are found to be consistent with a compact object merger origin, increasing the parallels between FXTs and GRBs. We finally discuss whether the source could originate in a higher redshift host galaxy.