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We present an observational study of the luminous red nova (LRN) AT\,2021biy in the nearby galaxy NGC\,4631. The field of the object was routinely imaged during the pre-eruptive stage by synoptic surveys, but the transient was detected only at a few epochs from $\sim 231$\,days before maximum brightness. The LRN outburst was monitored with unprecedented cadence both photometrically and spectroscopically. AT\,2021biy shows a short-duration blue peak, with a bolometric luminosity of $\sim 1.6 \times 10^{41}$\,erg\,s$^{-1}$, followed by the longest plateau among LRNe to date, with a duration of 210\,days. A late-time hump in the light curve was also observed, possibly produced by a shell-shell collision. AT\,2021biy exhibits the typical spectral evolution of LRNe. Early-time spectra are characterised by a blue continuum and prominent H emission lines. Then, the continuum becomes redder, resembling that of a K-type star with a forest of metal absorption lines during the plateau phase. Finally, late-time spectra show a very red continuum ($T_{\mathrm{BB}} \approx 2050$ K) with molecular features (e.g., TiO) resembling those of M-type stars. Spectropolarimetric analysis indicates that AT\,2021biy has local dust properties similar to those of V838\,Mon in the Milky Way Galaxy. Inspection of archival {\it Hubble Space Telescope} data taken on 2003 August 3 reveals a $\sim 20$\,\msun\ progenitor candidate with log\,$(L/{\rm L}_{\odot}) = 5.0$\,dex and $T_{\rm{eff}} = 5900$\,K at solar metallicity. The above luminosity and colour match those of a luminous yellow supergiant. Most likely, this source is a close binary, with a 17--24\,\msun\ primary component.