In the following model specification, which is a random intercept 2-level logistic regression:
Would two lower level units ($i$) with the same value of $x_{1ij}$ and within the same higher level unit ($j$) but with different values for $y_{ij}$ have the same predicted $\pi_{ij}$?
Yes they would and it has nothing to do with the model being mixed or logistic or whatever. In any regression, the predicted value for a unit depends on all the independent variables (x) and not at all on the dependent variable (y). That's why models aren't perfect.
For example, suppose your mixed model is students inside classrooms; suppose you have some student level variables and some classroom level variables; suppose you are predicting a passing grade. If two students have the same student level variables and are in the same class, the predicted probability of passing will be the same. Nevertheless, one may pass and one fail.
