The Difference of Definitions of "Vector" and "Tensor" in Mathematics and Physics

When I studied linear algebra in the mathematics department and electromagnetism in the physics department, I noticed that the definition of a vector varies significantly between the two disciplines. In linear algebra, a vector is a generalized concept of traditional vectors in Euclidean spaces. However, in electromagnetism, the definition of a vector is more stringent than that of traditional vectors. It's not uncommon for the same object to have different definitions in physics and mathematics, but typically, the definition of a vector in physics aligns with its mathematical counterpart. Which definition of a vector does physics use? Is it the one used in quantum mechanics or electromagnetism?
This issue perplexed me until I studied differential geometry. Differential geometry primarily studies manifolds, generalized forms of Euclidean spaces. Since manifolds may not necessarily be vector spaces, the most intuitive approach to consider vectors on them is through tangent spaces of manifolds, which are referred to as vector fields on manifolds.
Moreover, the spaces of interest in physics are often manifolds. Therefore, when physicists refer to vectors, they are actually referring to vector fields on the corresponding manifolds. Furthermore, in physics, the term "vector" does not denote a traditional vector in mathematics but rather an object known as a "vector field on a manifold."
Any space with coordinates can be viewed as a manifold. In other words, differential geometry applies to any space with coordinates. Nearly all coordinate-based spaces in physics necessitate differential geometry. While physics mathematics covers topics related to Euclidean spaces, the concept of manifolds is typically omitted. Therefore, many concepts remain unclear, including the differing definitions of vectors. Differential geometry helps us understand topics that are often overlooked in mathematics oriented towards physics and is crucial in nearly every aspect of physics.
Given the importance of differential geometry, theoretical physics students should acquire a fundamental understanding of it.