Update machine learning model while retaining the prediction results exact for old data
The safest way to keep the same predictions for your older data is versioning your trained models. It is, keeping the generated model artifact (in python could be pickle file, h5 file, etc) to make sure that you can use it getting the same results as you say, and generating new models (so new artifacts) via retraining when new data come in.
The usual approach is deciding an evaluation metric for your ML model, and based on it, when new data come in, you retrain your models (the frequency for this is something to decide on your team) and when the metric value is better (enough) than the latest model version, you can promote this new model for use in production. Theere are quite goos frameworks for models versioning, as MLflow.
Keep in mind that transfer learning does not guarantee predictions on older data are going to be same, since a new model is trained based on another model, but a different model is eventually created.