Is the vertical stream cropped from the wide one?

No. It is laid out for 9:16, so nobody is cut in half and every name fits.

Do I need a second OBS for vertical?

You need a vertical canvas: a second OBS profile at 1080 × 1920, or a vertical canvas plugin.

Help · Platforms

How to stream vertically for TikTok Live

atme makes a vertical version of your show for TikTok Live and Shorts, laid out for 9:16 instead of cropped: one person fills the frame, two stack, and three or more put whoever is talking on top. Copy the link under Settings → OBS → "Vertical, for TikTok Live" and add it as a 1080 × 1920 browser source on a vertical canvas in OBS. Streaming to TikTok from OBS needs a TikTok stream key, which TikTok gives to eligible accounts.

Settings → OBS → Copy link next to "Vertical, for TikTok Live"
In OBS, on a vertical canvas: Sources → + → Browser
Paste the link at 1080 × 1920

Shared screens and cards

A shared screen or Chatroulette goes across the top with the people under it; Starting, Be right back and Ending fill the frame.

Questions

Is the vertical stream cropped from the wide one?: No. It is laid out for 9:16, so nobody is cut in half and every name fits.
Do I need a second OBS for vertical?: You need a vertical canvas: a second OBS profile at 1080 × 1920, or a vertical canvas plugin.