The post-main-sequence evolution of massive stars encompasses phases in which the stars display high variability. One such class of objects are the blue supergiants. These objects may be in either the pre- or post-red-supergiant phase of their evolution. Their variability patterns might provide constraints for a proper classification of the objects. The study aims to characterise the observed variability of the B-type supergiant HD 14134 and to investigate the imprint of a time-variable wind on the brightness variation of the star and its impact on the detectability of pulsation signals. Spectroscopic data were collected over a five-month period and combined with space-photometry from the TESS mission. Stellar parameters were derived from a modelling of the time-averaged spectrum with CMFGEN and the spectral energy distribution and were confirmed with stellar evolution models computed with MESA. The light curves and radial-velocity curves of selected lines were analysed with various methods to determine pulsation signals. The wind variability and its imprint on the stellar brightness were investigated from an analysis of the Hα line. Predictions of mode excitations were computed with the GYRE pulsation code and compared to the frequencies determined from the observations. A g mode with a period of ∼ 19.2,d and its harmonics are consistently detected in all datasets. The spectra unveil strong, non-periodic wind variability, and three frequency signals were identified as being due to this wind variability. The stellar parameters and age derived for HD 14134 together with the absence of radial pulsations classify the star as a post-main-sequence object evolving towards the red-supergiant stage, putting into question its classification as a α Cyg variable. The results reinforce that simultaneous long-term spectroscopic and photometric monitoring is indispensable for reliable frequency detections and for disentangling variabilities imprinted by a time-variable wind from those imposed by pulsations.