Do new iPhones (iOS 13) support NFC in peer-to-peer mode?
No.  

Peer-to-Peer mode (like sharing contacts, URLs or pictures) is currently not on the road map.  

A quick look at the Apple developer site and we can confirm this.  The only function Core NFC supports (at the time of this writing), is tag reading and writing.


  Detect NFC tags, read messages that contain NDEF data, and save data
  to writable tags.


A Computerworld article, WWDC: Apple’s NFC plan is a big developer opportunity, seems to confirm this.  The focus appears to be reading/writing tags that enhance "guest experience" like that of payments, security, loyalty cards, and even drivers licenses and passports.  There's no mention of "convenience" like that of sharing digital assets between users.