Can random effects apply only to categorical variables?

This questions might sound stupid, but... is it correct that random effects could apply only to categorical variables (like individual id, population id, ...), e.g. say $x_i$ is categorical variable:

$y_i$ ~ $\beta_{x_i}$

$\beta_{x_i}$ ~ $Norm(\mu, \delta^2)$

but from the principle the random effect cannot apply to continuous variable (like height, mass...), say $z_i$:

$y_i$ ~ $\alpha + \beta \cdot z_{i}$

because then there is only one coefficient $\beta$ which cannot be constrained? Sounds logical but I wonder why it is never mentioned in statistical literature! Thanks!

EDIT: But what if I constrain $z_i$ like $z_i$ ~ $Norm(\mu, \delta^2)$? Is it then random effect? But this is different from the constrain I put on $\beta_{x_i}$ - here I constrain the variable whereas in the previous example I constrained the coefficient! It starts to look as a big mess to me... Anyway, it doesn't make much sense to put this constraint, because $z_i$ are known values, so maybe this idea is completely odd :-)

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