Biofabrication is providing scientists and clinicians the ability to produce engineered tissues with desired shapes, and gradients of composition and biological cues. Typical resolutions achieved with extrusion-based bioprinting are at the macroscopic level. However, for capturing the fibrillar nature of the extracellular matrix (ECM), it is necessary to arrange ECM components at smaller scales, down to the micron and the molecular level. Herein we introduce a bioink containing hyaluronan (HA) as tyramine derivative (THA) and collagen type 1 (Col 1). In this bioink, similarly to connective tissues, Col is present in fibrillar form and HA as viscoelastic space filler. THA was enzymatically crosslinked under mild conditions allowing simultaneous Col fibrillogenesis, thus achieving a homogeneous distribution of Col fibrils within the viscoelastic HA-based matrix. THA-Col composite displayed synergistic properties in terms of storage modulus and shear-thinning, translating into good printability. Shear-induced alignment of the Col fibrils along the printing direction was achieved and quantified via immunofluorescence and second harmonic generation. Cell-free and cell-laden constructs were printed and characterized, analyzing the influence of the controlled microscopic anisotropy on human bone marrow derived mesenchymal stromal cells (hMSC) migration. Anisotropic HA-Col showed cell instructive properties modulating hMSC adhesion, morphology and migration from micro pellets stimulated by the presence and the orientation of Col fibers. Actin filament staining showed that hMSCs embedded into aligned constructs displayed increased cytoskeleton alignment along the fibril direction. Based on gene expression of cartilage/bone markers and ECM production, hMSCs embedded into the isotropic bioink displayed chondrogenic differentiation comparable to standard pellet culture by means of proteoglycan production (Safranin O staining and proteoglycan quantification). The possibility of printing matrix components with control over microscopic alignment brings biofabrication one step closer to capturing the complexity of native tissues.