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).
After some research (thanks Robert for your input), I've found that:
No, you can't tweak your own eSIM to have the same number as other SIMs or eSIMs of yours.
However, it is possible to have the same number on multiple devices if your provider supports it (you will most likely have to pay the extra cost of such feature).
For example, in the UK, where I live, as of today (30th October 2020), only two operators offer that:

EE (Smart Number --only Apple devices supported)
Vodafone (One Number --only Apple and Samsung devices supported)

EDIT: Also O2 appears to offer this feature, although it doesn't seem to have a branded name and I've only found a related section on their website about Samsung.