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 V_sd. 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. We attribute their similar low-energy behavior to similar interaction-enhanced spin-fluctuations in regions of low charge density that 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.