The way to represent models can vary a lot with the application, and I'm not sure if there is any name rather than "vector" form or "observation level" form.
We need to know if we are numbering the individuals within groups or not. This only makes sense if it was a designed study with blocks (one categorical variable) and a treatment factor (a second categorical variable), or some other formulation, such as in a repeated measurements study.
But one way I would write is the following, together with the rationale behind it:
First, we note that we have an intercept in the formulation. Second, there are two categorical variables, factor 1 and factor 2. Meaning we will need two indexed parameters representing them, and two indices are need to correctly identify an observation. Third, we have two measurements at the individual level that are combined via splines.
Let us assume we have numbered the individuals/observations regardless of the group. So we have the index of individuals as $i = 1,...,n$. Also let $j = 1, ..., m$ be the factors of the first categorical variable and $k = 1, ..., l$ be the factors of the second categorical variable. Lastly, let $x_i$ and $z_i$ be the value of covariate $x$ and $z$ for individual $i$. If we model, initially, that the continuous variables linearly and with interaction taken into account, we can write it like:
$$
\text{q}_{0.5}(Y_{ijk}) = \alpha + \gamma_{j} + \delta_{k} + \beta_1 x_i + \beta_2 z_i + \beta_3 x_i z_i
$$
Where $i = 1,...,n$, $k = 1, ..., l$ and $j = 1, ..., m$. In order to have the model identifiable, we need to choose which factors from the first and second categorical variables to be the reference level. So, we have that $\gamma_1 = \delta_1 = 0$ just in order to have the model identifiable.
If we model the continuous variables as splines, assuming we have only one function for every individual (it doesn't vary or "interact" with another group) we can have the following formulation, as proposed in your question:
$$
\text{q}_{0.5}(Y_{ijk}) = \alpha + \gamma_{j} + \delta_{k} + \text{f}_1(x_i) + \text{f}_2(z_i) + \text{t}(x_i, z_i)
$$
Still using the same restrictions $\gamma_1 = \delta_1 = 0$.
To add an interaction between the first and second categorical variables, we are essentially adding another parameter representing it, ending up with:
$$
\text{q}_{0.5}(Y_{ijk}) = \alpha + \gamma_{j} + \delta_{k} + \Delta_{jk} + \text{f}_1(x_i) + \text{f}_2(z_i) + \text{t}(x_i, z_i)
$$
