(abridged) Powerful radio galaxies often display enhanced optical/UV emission regions, elongated and aligned with the radio jet axis. The aim of this series of papers is to separately investigate the effects of radio power and redshift on the alignment effect, together with other radio galaxy properties. In this second paper, we present a deeper analysis of the morphological properties of these systems, including both the host galaxies and their surrounding aligned emission. The host galaxies of our 6C subsample are well described as de Vaucouleurs ellipticals, with typical scale sizes of ~10kpc. This is comparable to the host galaxies of low-z radio sources of similar powers, and also the more powerful 3CR sources at the same redshift. The contribution of nuclear point source emission is also comparable, regardless of radio power. The 6C alignment effect is remarkably similar to that seen around more powerful 3CR sources at the same redshift in terms of extent and degree of alignment with the radio source axis, although it is generally less luminous. The bright, knotty features observed in the case of the z~1 3CR sources are far less frequent in our 6C subsample; neither do we observe such strong evidence for evolution in the strength of the alignment effect with radio source size/age. However, we do find a very strong link between the most extreme alignment effects and emission line region properties indicative of shocks, regardless of source size/age or power. In general, the 6C alignment effect is still considerably stronger than that seen around lower redshift galaxies of similar radio powers. (abridged)