A tensor is a mathematical object which has to obey certain rules about how to transform it from one coordinate system to another.
Engineers started using and measuring strains a century or more before tensors were invented (by Ricci, in around 1900, and not in the context of continuum mechanics). They just happened to make a bad choice of how to measure shear strains (though they got it right for shear stresses).
You can create an "engineering formulation" of continuum mechanics that doesn't use tensors at all, by pretending the 6 stress components are a "vector," using matrix arithmetic in an ad hoc way, and inventing complicated-looking formulas for transforming stresses and strains into different coordinate systems.
Alternatively, you can change the definition of shear strain by a factor of two, and use mathematics that doesn't need any "special" definitions, just standard vector calculus.
The second way has obvious advantages if you want to combine continuum mechanics with other phenomena such as fluid dynamics, or with special or general relativity.
Really, this is no more of an issue than measuring angles in degrees in real life, or in radians for mathematical purposes.