It is a vector that doesn't follow vector laws.

A quantity that doesn't obey vector laws cannot be considered a vector.

Tension (or compression) in general is not a force but a pair of equal and opposite forces acting at a certain point. In problems in Physics and engineering, we often consider only one of the two forces in the tension pair and call it tension or tensile force. In this way, we can consider tension as a vector.

When you go by the layperson's definition of a tensor in Physics that says, "tensor is a physical quantity which has one magnitude and multiple directions.", you may consider tension as a tensor. However, tension does not fit the mathematical matrix in order to be a mathematical tensor. This is why the layperson's definition of a tensor isn't a valid one.

I hope my explanation helps.

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