In usual multiple regression the response variable $y$ is 1-dimensional so for each sample we can write an equation $$y = \boldsymbol \beta^\top \mathbf x + \epsilon,\tag{1a}$$ where $\mathbf x$ is an $r$-dimensional vector of predictors and $\boldsymbol\beta$ is the vector of regression coefficients. If the sample size is $n$, we can combine $n$ such equations into one, stacking all $y_i$ into one vector $\mathbf y$ and all $\mathbf x_i$, as rows, into one data matrix $\mathbf X$. This yields the form that you gave as Equation 1: $$\mathbf y = \mathbf X\boldsymbol\beta + \boldsymbol\epsilon.\tag{1b}$$

Your Equation 2 describes a single sample of multivariate regression where response variable $\mathbf y$ is a vector. Using the similar notation as above, we could write for each sample $$\mathbf y = \mathbf B\mathbf x + \boldsymbol\epsilon,\tag{2a}$$ with the only difference to your Equation 2 being that here the intercept is included in $\mathbf x$ (i.e. the first element of $\mathbf x$ is always equal to $1$) and so $\mathbf B = [\boldsymbol \mu, \mathbf C]$ stacked together. Here we can also combine $n$ such equations together by using data matrices: $$\mathbf Y = \mathbf {XB} + \mathbf E.\tag{2b}$$

The key point and probably the source of confusion for you was that $\mathbf y$ in Equations (1b) and (2a) above are very different things! In (1b) it denotes an $n$-dimensional data vector ($n$ one-dimensional sample points) while in (2a) it denotes an $s$-dimensional response vector (one $s$-dimensional sample point).

In usual multiple regression the response variable $y$ is 1-dimensional so for each sample we can write an equation $$y = \boldsymbol \beta^\top \mathbf x + \epsilon,\tag{1a}$$ where $\mathbf x$ is an $r$-dimensional vector of predictors and $\boldsymbol\beta$ is the vector of regression coefficients. If the sample size is $n$, we can combine $n$ such equations into one, stacking all $y_i$ into one vector $\mathbf y$ and all $\mathbf x_i$, as rows, into one data matrix $\mathbf X$. This yields the form that you gave as Equation 1: $$\mathbf y = \mathbf X\boldsymbol\beta + \boldsymbol\epsilon.\tag{1b}$$

Your Equation 2 describes a single sample of multivariate regression where response variable $\mathbf y$ is a vector. Using the similar notation as above, we could write for each sample $$\mathbf y = \mathbf B^\top\mathbf x + \boldsymbol\epsilon,\tag{2a}$$ with the only difference to your Equation 2 being that here the intercept is included in $\mathbf x$, i.e. the first element of $\mathbf x$ is always equal to $1$, and so $\mathbf B = [\boldsymbol \mu, \mathbf C^\top]$ stacked together. Here we can also combine $n$ such equations together by using data matrices: $$\mathbf Y = \mathbf {XB} + \mathbf E.\tag{2b}$$

The key point and probably the source of confusion for you was that $\mathbf y$ in Equations (1b) and (2a) above are very different things! In (1b) it denotes an $n$-dimensional data vector comprising $n$ one-dimensional sample points while in (2a) it denotes an $s$-dimensional response vector which is one single $s$-dimensional sample point.