I have $N$ data points where each is a 2-D matrix $\mathbf{Y} \in \mathbb{R}^{d \times*}$ which means that they are different in their second dimension. I want to use a notation to recall them all together, for instance as an input argument to a function or an algorithm.
I may use $\{\mathbf{Y}^i\}_{i=1}^N$ to show I mean all of them and $\mathbf{Y}^i$ to denote the $i$th matrix. But, I prefer a more condensed notation. Otherwise, I need to use it for example like $$K=f(\{\mathbf{Y}^i\}_{i=1}^N)$$ which may not look so nice!
Use ${\bf Y}$ for your full data set. This ${\bf Y}$ has $N$ components $Y_i$ $(1\leq i\leq N)$ corresponding to the $N$ data points, each of these being a matrix $Y_i\in{\mathbb R}^{d\times n_i}$. Then you can talk about $K=f({\bf Y})$.