Blood vessels are efficient living architectures built up via growth and remodelling processes finely orchestrated by cascades of chemo-mechanical events that balance tissue turnover and mechanical stress. Alterations in homeostasis can lead in fact to structural and cellular impairments potentially kindling many vascular diseases. In order to explore how mechanobiology concurs to steer vessel functions and ensure its integrity at any scale, there is the need to accurately characterize internal stress distributions in a way to identify stress-induced signalling that modulate microscopic activity of harbouring cell populations and overall tissue performance. Starting from some experimental evidences and tissue analogues suggesting that combined in vivo conditions would call into play complex microstructural arrangements with a high degree of anisotropy, an enriched microstruturally-motivated hyperelastic model is proposed to include the effects of the multilamellar organization of vessel layers, giving the possibility to extend the capabilities of largely adopted models that however seem to limit the constitutive class of vessel material, homogenization inevitably obscuring some complex phenomena that emerge when the hierarchy and the lamellar organization of vessel elastic tunicae are taken into proper account. In this respect, different loading and prestretch conditions will be analysed with the aim to gain both new insights into how the specific arrangement of vessel layers contributes to their effective mechanical response and cues helping to decipher the way in which the competition of tissue stresses orient events of vascular physiology and mechanotransduction through nonstandard stress-based mechanisms.