What you see

When you get the DropCard, the email has three parts:

A cover letter, if the sender wrote a note for you.
The contact information, inline
vCard with the contact information

What you can do and cannot do

View the VCF file in Gmail

Save the contact information

into Gmail's contacts

If you hold the mouse over the plus button, you will see "Add to Contacts"

Click the ("person and plus") button, and the contact will be saved to the Gmail contacts.

into Apple Contacts (on macOS)

Tap on the download (down arrow) button

Gmail/Chrome will download the card, but not trigger the helper app. (Not clear why, still investigating).

This will not make the card info available in Gmail, although the email address can be added to your "Other Contacts" in Gmail, if you reply to the email.

If you email someone but you haven't added them as a contact, Google Contacts automatically saves their email address in your "Other Contacts" group. Their email address will show up the next time you email them.
https://support.google.com/mail/answer/1069522?hl=en&rd=1

DropCard documentation