vCards is the preferred way of sharing contacts in Delta Chat now. When you select “Contact” in attach menu, vCard of the contact gets attached.
There is however no easy way to send your own vCard to someone, e.g. in another messenger. Workaround is to send a vCard of self-contact to self into Saved messages, then to “Export attachment”, then attach it somewhere.
I think a good place to have a “Share” button is in my own profile on the same screen where I can change my name and avatar, both on Android and Desktop. On Android this should probably share the vCard to other apps (rather than to chat) and on Desktop export a file.
There is a similar feature request about sharing the address, but we don’t really want users to share just an address without the key anymore because unencrypted messages are not accepted by chatmail relays: