Biobased and biodegradable polymeric materials are a sustainable alternative to the conventional plastics used in food packaging. This study investigated the possible effect of biobased cling films derived from renewable and circular and sustainable sources on key cheese sensory parameters (appearance and odor) able to influence consumer acceptance or rejection of a food product over time. For this purpose, a semi-hard cheese was selected as food model and stored for 14 days at 5 °C wrapped with five cling films: two bio-plastic materials from renewable circular and sustainable sources (R-BP1 and R-BP2), one bio-plastic film from a non-renewable source (NR-BP), and two conventional cling films (LDPE and PVC). Three analytical approaches (image analysis, electronic nose, and sensory test) were applied to evaluate the variation and the acceptability in terms of appearance and odor of the cheese. In preserving cheese color, the R-BP1 and RBP2 films were comparable to LDPE film, while NR-BP film was comparable to PVC film. In terms of odor preservation, R-BP2 film was comparable to LDPE and PVC. The consumer test showed that appearance and odor scores were higher for cheeses stored in R-BP1 and R-BP2 films than NR-BP film. Moreover, in terms of odor, R-BP1 film performed better than conventional films. This study shows how biodegradable cling films from renewable circular and sustainable resources could have comparable performance to conventional plastics (LDPE and PVC) used in the food sector.