People are free to use any notation they want. There is no "single objective" meaning of $\varepsilon$. If you want, you can as well write regression equation as $X = \varepsilon_0 + \varepsilon_1 Y_1 + \varepsilon_2 Y_2 + \beta$. To give other examples, $\pi$ is a number, but it is also commonly used to denote parameters of different models, e.g. parameters of latent class analysis models
$$ f(Y_i;\pi_r) = \prod_{j=1}^J \prod_{k=1}^{K_j} (\pi_{ijk})^{Y_{ijk}} $$
See other questions tagged notation to learn that notation is arbitrary.
All this said, it is worth emphasizing that you should be clear about the notation you are using (define the meaning of your symbols) and try to be consistent with the conventions in your field. If people in your field more commonly use $\varepsilon$, then use it, if they more commonly use $u$, use it instead. Notice however that there are cases when you want to use different notation, e.g. people commonly use $\beta_0,\beta_1,\dots$ for regression parameters, but if you already used $\beta$ symbols for something else, then it may be wiser to use other symbols not to confuse your reader. So in the end we want the notation to be flexible, the same as human language is flexible, because there are cases when we need to say something using different words/symbols.
