I was wondering if a saved model in a Pipeline object contains the score of the data with which it has been trained. If so, how to get that score without having to put the data back in?
This depends on the model. Mostly, no. Also, you'd need to specify what exactly you mean by "score"; there are many metrics that might be saved somewhere.
kNN models store the training data in a private attribute _fit_X (source), so you could recreate a score from that (though you're not really saving much work here).
HistGradientBoosting models store an iteration-wise training and validation score (docs). GradientBoosting models similarly save loss functions at each iteration.
Cross-validation models like LogisticRegressionCV save the cross-validation scores for each hyperparameter value. Those are significantly different from a training score though.