Aims: We characterise the underlying stellar host in a sample of 20 blue compact galaxies (BCGs), by fitting their two-dimensional light distributions. Their derived host structural parameters and those of eight other BCGs already obtained in a previous paper are related to galaxy properties, such as colours and gas content. These properties are also compared with those of other galaxy types. Methods: The structural parameters of the host were derived by fitting a two-dimensional PSF-convolved Sérsic model to deep optical images in several bandpasses (B, V, R, I). We followed a fitting technique that consists in the accurate masking-out of the star-forming regions in several steps. Results: All the BCG hosts but one show low Sérsic indexes (0.5⪉ n ⪉ 2), with mean effective radius < r_e,B > = 1.11 ±0.74 kpc and mean surface brightness < μ_e,B > = 22.59 ±0.68 mag arcsec-2. Host effective radii scale linearly with their luminosity, while n and μe do not. In addition, host colours and structural parameters are not linearly correlated. Overall, the flux enhancement caused by the starburst is about 0.8 mag, while their B-R colours decrease by about 0.2 mag. Galaxies with more luminous and extended hosts show larger and luminous starburst components, whereas the relative strength of the burst (L_burst/L_host) does not show any significant dependence on the host luminosity (or mass). While hosts show B-R = 0.95±0.26 in median, galaxies with redder hosts (< B-R > = 1.29± 0.10) and with bluer hosts (< B-R> = 0.66±0.10) are distinguished among the more and less luminous systems, respectively. Overall, BCG hosts are more compact (by a factor ~2) and have higher central surface brightnesses (by about ~2 mag) than dIs and most dEs. BCG hosts and isolated dIs are indistinguishable in the B-band Tully-Fisher relation (TFR). We found that about 50-60% of the galaxies are more underluminous than those late-type discs with the same circular velocity. This feature is more important when luminosities are converted into stellar masses, while it tends to diminish when the Hi gas mass is added. Deviations among host masses for a given circular velocity from the stellar TFR correlate with their Hi mass-to-luminosity ratio (M_HI/LB), whereas deviations from the gas+stellar TFR do not. Overall, our findings suggest that the baryonic mass in BCGs tends to normal values, but BCGs tend to be inefficient by producing stars, especially toward the low-mass, gas-rich, and bluest hosts, in a similar way to dIs. Table 4, Figs. 10 and 11 are only available in electronic form at http://www.aanda.org