The relationship between linear codes in the Hamming metric and projective algebraic varieties has led to deep interactions between coding theory and algebraic geometry, with classical examples such as Reed-Solomon codes and the rational normal curve. On the other hand, the sum-rank metric has recently gained attention due to applications in network coding, distributed storage, and post-quantum cryptography, with linearized Reed-Solomon codes emerging as optimal constructions. Despite recent advances, their structural and geometric properties are still not fully understood, and existing distinguishers remain limited. In this paper, we develop a geometric framework for linearized Reed-Solomon codes by considering a $q$-analogue of the rational normal curve. This yields a geometric characterization for certain parameter choices and reveals that the corresponding sets of points satisfy unexpectedly many $(q+1)$-degree hypersurface conditions. Our approach extends Schur-product-based techniques from the Hamming and rank-metric settings to the sum-rank metric case. Finally, we study the Hilbert function of the associated coordinate ring, providing a detailed description of its behavior and identifying its regularity, which also sheds new light on Gabidulin codes.