To the best of my knowledge the operations in deep learning are very much tensor operations. In most cases they seem to boil down to matrix multiplicaiton on one of the indices, but it's still kind of useful to use tensor notation (especially using the Einstein summation convention) since IMHO T_abc M_ax = S_xbc is easier to read than the corresponding sum, or the corresponding pseudocode.

Not sure if the fact that tensors have slightly more structure than mere arrays w.r.t. e.g. a change of basis is that useful, but it could be useful in proofs, especially if a FFT is involved somewhere.