No, it does not make sense to do this.
You model has learned how to map one input space to another, that is to say it is itself function approximation, and will likely not know what to for the unseen data.
By not performing the same scaling on the test data, you are introducing systematic errors in the model. This was pointed out in the comments by nanoman - see that comment for a simple transformation example.
To exaggerate the point, imagine you have a language model, translating text from English to French. You apply a single transformation on your English data: you first translate it to Spanish. Now the model is trained to translate Spanish to French, with some accuracy. Now you move on to the test data - still in English - and you do not apply the same transformation as you did to your training data. You are asking the model to translate directly from English to French, instead of Spanish to French, and it is obvious that the results won't be good.
In principal, this idea is the same as with any other model and transformations, just that the impact might not always be so visible i.e. you might get really lucky and not notice a large impact.
The language model might have learnt some elementary linguistics common to all three languages (e.g. overlapping vocabulary or sentence structuring), but we cannot expect the model to perform well, translating English to French.
You should only compute the transformation statistics (e.g. mean and variance for normalisation) only on training data and use these values to then transform the training data itself, and then the same values to transform the test data.
Including the test dataset in the transform computation will allow information to flow from the test data to the train data and therefore to the model that learns from it, thus allowing the model to cheat (introducing a bias).
Also, it is important not to confuse transformations with augmentations. Some "transformations" might be used to synthetically create more training data, but don't have to be used at test time. For example, in computer vision, deleting regions of an image. Test time augmentation is something you could read about.
More complicated models (ones with many many more parameters) might be able to perform some kind of interpolation, especially if your dataset if N-dimensions with a large N (e.g. > 10).
This has recently been seen with extremely large models, such as Open AI's GPT-3, which has 175 BILLION parameters, and is therefore even able to perform quite well on completely different tasks, let alone the given problem in the training set range.