I suppose what surprises me is that the default position is to have customers contact you separately about changing their email address, this feels inherently less secure.
Shopify is a fantastic platform, but this kind of basic functionality should be included.
@JL_8 - Shopify have made no announcements or plans to build this in natively.
Shopify has recently launched new customer accounts. If you are using new customer accounts, customers can now change their name but NOT their email address. See screenshot: https://share.heliumdev.com/z8uRx7oO
There is currently NO way to edit customer email via new customer accounts, but it is possible via classic customer accounts although an app is required.