Is Geometric Algebra isomorphic to Tensor Algebra?

Every element of a geometric algebra can be identified with a tensor, but not every tensor can be identified with an element of a geometric algebra.

It's helpful to consider the vector derivative of a linear operator, of a map from vectors from vectors. Call such a map $\underline A$. The vector derivative is then

$$\partial_a \underline A(a) = \partial_a \cdot \underline A(a) + \partial_a \wedge \underline A(a) = T + B$$

where $T$ is a scalar, the trace, and $B$ is a bivector. The linear map $\underline A$ can then be written as

$$\underline A(a) = \frac{T}{n} a + \frac{1}{2} a \cdot B + \underline S(a)$$

where $\underline S$ is some traceless, symmetric map. While the scalar can be turned into a multiple of the identity, in $T \underline I/n$, and the bivector can be directly turned into an antisymmetric map in $a \cdot B$, the map $\underline S$ is very much part of $\underline A$, yet not representable in general through a single algebraic element of the geometric algebra. This is just one example of such an object.

A tensor (not an abstract tensor in the sense of Penrose) is a multilinear function from the repeated Cartesian product of a vector space the real numbers thus the multi-linear function $T(a_1,...,a_r)$ where $a_1$ through $a_r$ are vectors in the same vector space. The rank of this tensor would be $r$. All the standard tensor operations can be defined without regard to component indices. Go to GA and look at bookGA.pdf in "GA Notes". Note that any tensor has a definite rank (number of multi-linear vector arguments), but a multivector can be the sum of different pure grade multivectors. An even mulitvector (contains only even grade components) can represent a spinor, but a tensor cannot. In that sense a multivector is more general than a tensor. However, pure grade multivectors can only represent completely antisymmetric tensors. In that sense tensors are more general than pure grade multivectors. The rank of a tensor is not restricted by the dimension of the base vector space like the grade of a multivector.

Similar question: what's the relationship of tensor and multivector

The short answer is that all multivectors are tensors, but not all tensors are multivectors, so geometric algebra is not and cannot be isomorphic to tensor algebra.

However, we can try to embed geometric algebra inside of tensor algebra. These resources give some guidelines for doing so (in the special case where the vector space is $\mathbb{R}^3$ and the inner product is the "regular" inner product, as opposed to say the Minkowski metric on $\mathbb{R}^4$):

http://www2.ic.uff.br/~laffernandes/teaching/2013.1/topicos_ag/lecture_18%20-%20Tensor%20Representation.pdf

https://www.docdroid.net/uwfvUxE/tensor-representation-of-geometric-algebra.pdf.html

Tensor algebras are (in general) infinite dimensional, but the most common Clifford algebras (over finite dimensional vector spaces) are finite dimensional, so they aren't isomorphic.

Clifford algebras can be constructed as quotients of tensor algebras, so there is a relationship between them.