I've come across the following operation. It outputs a vector of length $n$ with elements $$\left[\vec x^T M_i \; \vec x\right]$$ where $\vec x$ is a vector of length $n$, $\vec x^T$ is its transpose, $M_i$ is a symmetric $n \times n$ matrix, and $i = 1, 2, \ldots, n$.
The $M_i$ matrices form a 3-dimensional $n \times n \times n$ array. Is this array a tensor? If so, of what kind? And what would be the appropriate notation? (I'm new to tensors, so any general pointers would be greatly appreciated.)
Computer scientists tend to call any multidimensional array a tensor. So, in that sense, it is a tensor. You can write its components as $M_{i}^{jk}$. Then, say, write the output vector as $v^i=M^i_{jk}x^jx^k$.
However, mathematically speaking, a tensor is expected to follow specific transformation law (e.g. see here), using the Jacobian of a coordinate transformation. It's not really clear how this makes sense for your fixed matrices. The basic idea is that tensors represent geometric objects that have meaning independent of the coordinate system.
For instance, it's kind of like asking whether some random matrix can be considered a Cauchy stress tensor... sure it can, but to make this conclusion requires some background assumptions about the physical system it describes.