What's going on here is a collision between the way certain terms are used in two different fields.

People who learn about tensors and vectors in the culture of Mechanical Engineering, Applied Mathematics, or Physics, are strenuously trained that a "vector" is not just any list of numbers, but a specific mathematical object with specific properties. (Similarly with tensors.) In that culture, the use of the term "vector" the way it is used in Software Engineering looks quite odd. An undergraduate might even have a problem set that shows various number triples and be asked to identify which ones are vectors and which ones aren't.

I was educated under that system, but spent years as a software engineer, so I am comfortable switching between the two contexts. When I wear my Software Engineer hat, I am quite comfortable with the generalized use of vector and tensor as done in software and ML, but for someone who has only grown up in one culture, it looks very wrong.

Given a vector space V over a field F, a tensor is a multilinear map

   T: V x V x V x ... x V^* x V^* x V^* -> F

You're mixing up geometric objects (multilinear transformations) with their coordinate representations. A geometric object, such as a multilinear map, is automatically invariant to the coordinate system. An array of numbers, like a matrix, is not.