Do I need to enter participants email or phone numbers?
No. You only type the names. The WhatsApp message goes out through your own app: the button opens the contact picker with the text ready, and no number goes through the tool.
Can someone guess the match from the link?
At a glance, no: the name does not appear as readable text. But whoever has the link and knows the format can decode it, because there is no password. That is why a person link should never be posted in a group.
Does Google Analytics see the fragment?
The tool strips the fragment from the address bar and from the page address it reports to Analytics as soon as it reads it, and no measurement event carries names. Even so, the original link stays in the message of whoever received it, which is outside the site control.