We address the question whether GRB 111209A was a special event beyond its extreme duration alone, and whether it is a classical GRB or another kind of high-energy transient. Furthermore, we place SN 2011kl into the context of large samples of SNe, addressing in more detail the question of whether it could be radioactively powered. We present afterglow photometry obtained in seven bands with the GROND imager as well as in further seven bands with the UVOT telescope on-board \emph{Swift}. The light curve is analysed by multi-band modelling and joint fitting with power-laws and broken power-laws. We model SN 2011kl using SN 1998bw as a template and derive a bolometric light curve including near-infrared data. We compare the optical afterglow and the properties of SN 2011kl to large ensembles we have analysed in earlier works, additional GRB-SNe analysed here, as well as literature results on stripped-envelope and superluminous supernovae. We find a strong, chromatic rebrightening event at $≈0.8$ days after the GRB. After this, the light curve decays achromatically, with evidence for a break at about 9 days after the trigger. The afterglow luminosity is found to not be exceptional. Comparison in the $k,s$ context, i.e., comparing it to SN 1998bw templates in terms of luminosity and light-curve stretch, clearly shows SN 2011kl is the most luminous GRB-SN to date, and it is spectrally very dissimilar to other events, being significantly bluer/hotter. SN 2011kl does not reach the luminosities of true superluminous supernovae, though, and evolves faster than any of them. (abridged) ; Comment: A&A, submitted; 50 pages: 22 pages article, 15 pages appendix, 13 pages tables