Thank you @bfarber for your quick reply on this subject.
From my point of view, being able to know who is the customer is mandatory in Commerce. 😉
If that could help during the internal discussions ...
I wanted the last name of the customer in order to print labels (full address) to put on the enveloppes when sending products, for instance.
But also to track recurring purchases in order to send gift to those having purchased more than x times a certain product over the years.
The idea is to automatise the whole process thanks to the Rest API.