After I instantiate a scikit model (e.g. LinearRegression), if I call its fit() method multiple times (with different X and y
If you will execute model.fit(X_train, y_train) for a second time - it'll overwrite all previously fitted coefficients, weights, intercept (bias), etc.
If you want to fit just a portion of your data set and then to improve your model by fitting a new data, then you can use estimators, supporting "Incremental learning" (those, that implement partial_fit() method)