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Let $V$ be a set of vectors of length $n$. Define a $k$-fold product on $V$, $$ \Upsilon(\{v_1,\ldots,v_k\}):=\sum_{j=1}^n\prod_{i=1}^k v_{ij}, $$ where $v_i\in V$ and $v_{ij}$ is the $j^\text{th}$ element of $v_i$. So $$ \Upsilon(\{\})=n $$ is the dimension of the vectors in $V$, $$ \Upsilon(\{v\})=\text{sum of elements of $v$}, $$ and $$ \Upsilon(\{v,w\})=v\cdot w. $$

Is there a standard name or notation for this product?

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I don't know of a standard notation or even a name for this product, though it arises in physics quite naturally, among many other places. In fact, we can view it as the discrete analogue of an integrated product of functions. I choose the index convention $\Upsilon(\{v^1,\ldots,v^k\})=\sum_{i=1}^n\prod_{j=1}^k v^j_i$.

One way to think of $\Upsilon$ is to consider the tensor $$T_{i_1\cdots i_k} = v^1_{i_1}\cdots v^k_{i_k}$$ where the indices $i_j$ range from $1$ to $n$. Then the product is the contraction of $T$ with the tensor $$C^{i_1\cdots i_k} = \begin{cases} 1, & \textrm{if }i_1=\ldots=i_k \\ 0, & \textrm{else}. \end{cases}$$ The product would then be written as $$\Upsilon(\{v^1,\ldots,v^k\}) = C^{i_1\cdots i_k}v^1_{i_1}\cdots v^k_{i_k}.$$ For "small products," perhaps the notation $v\cdot w\cdot x$ or $v_i w_i x_i$ could be used, where summation is implied. For large products, consider the shorthand $\prod_j v^j_i$ or $v^1_i\cdots v^k_i$.

Notice that the product is the analogue of $$\begin{equation*} \int d x\, \phi^1(x)\cdots \phi^k(x).\tag{1} \end{equation*}$$ (Careful, the upper indices here are just labels.) Products of the form $\int d x\, \phi(x)^4$ arise naturally in interacting quantum field theories. The "child's version" of this product, used in toy models of quantum field theories where spacetime has been made discrete${}^\dagger$, is the sum $\sum_{i=1}^n \phi_i^4$, or $\Upsilon(\{\phi,\phi,\phi,\phi\})$. This is often written in shorthand as $\phi_i^4$ or even $\phi^4$. The notation can be confusing since $\phi$ may carry indices other than spacetime indices. You might see the product $(\phi\cdot \phi)^2$ where the dot product is over some "internal" space and then the whole thing is summed over the spacetime indices, $(\phi\cdot \phi)^2 = \sum_i(\sum_a\phi_i^a\phi_i^a)^2$. Of course, this can be written using $\Upsilon$, but the expression is unwieldy compared to the shorthand used here.

${}^\dagger$ If we fold (1) with $\sum_{i=1}^n \delta(x-x_i)$ we get $\Upsilon(\{ \phi^1, \ldots, \phi^k \}) = \sum_{i=1}^n \prod_{j=1}^k \phi^j(x_i)$.

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