Quantum point contacts and quantum dots, two elementary building blocks of semiconducting nanodevices, both exhibit famously anomalous conductance features at low energy scales: the 0.7-anomaly in the former case, the Kondo effect in the latter. For both effects the conductance shows a remarkably similar low-energy dependence on temperature (T), magnetic field (B) and source-drain voltage. This has led to the suggestion that the 0.7-anomaly and the Kondo effect have the same microscopic origin. Here we explore this notion theoretically and experimentally by studying the geometric crossover between a quantum dot and a quantum point contact. We introduce a one-dimensional model that reproduces the essential features of the geometry- and B-dependence of the conductance at T=0 for both Kondo effect and 0.7-anomaly. Though their properties differ markedly at high energies, at low energies there are striking similarities. We attribute the latter to similar interaction-enhanced spin-fluctuations in regions of low charge density and conjecture that these can be described using similar Fermi-liquid theories. Our predictions are consistent with our experimental results, which confirm, in particular, that the 0.7-anomaly exhibits Fermi-liquid behavior at low B and T. We also explain in detail how the 0.7-structure at T=B=0 arises from a combination of geometry and interaction effects.