Part of the regression model entails modeling the conditional variance of Y given your predictors x; Var(Y|x). In order to scale the Y variable to unit variance before fitting the model you would have to use the marginal variance Var(Y). This marginal variance is of no importance to the regression model however.
In case you use an error function location-scale from the location scale family such as the Gaussian, and assume homoscedasticity, scaling the response variable does not affect the final model. However, when the variance depends on the mean, such as for count data, scaling the response variable prior to model fitting distorts the mean-variance relationship.
Scaling of the regressors may be useful to eliminate effects of the units used and enable direct comparison of their coefficients, but when using only univariate regression this argument does not hold for the response variable.
In conclusion, scaling the response variable only complicates interpretation, and is outright wrong if the variance of Y is not constant.