Is it possible to "mirror" my smartphone SIM into my smartwatch eSIM?

I am wondering whether, with eSIM technology, I would be able to mirror my smartphone physical SIM to my smartwatch eSIM (as if I had cloned my phone, like in the movies). Telling you my final goal could explain better what I am trying to achieve. Basically I am trying to understand if I could leave my smartphone at home (SIM included), then going out with my smartwatch (e.g. running/cycling/etc) and still receive my smartphone's notifications, texts, etc from the 3G/4G network (essentially I'll be out of bluetooth range).