To clear things up, I have Fizban’s Treasury of Dragons, but when I was looking at selection for dragon ancestry in character creator, I couldn’t find gem options. Please help.

Assuming you own the full book and not just the compendium version, it should be under Dragonborn:

Make sure your entitlements are in sync here too: https://www.dndbeyond.com/account/marketplace

My dungeon master bought the book and he has the Master subscription. But I don’t have the option to select subraces of Dragonborn like in that photo. And how do I know it is not the compendium version?

Quote from GlareMaster >>

My dungeon master bought the book and he has the Master subscription. But I don’t have the option to select subraces of Dragonborn like in that photo. And how do I know it is not the compendium version?

 Is the character in question in the campaign your DM Has created? If not, that’s your issue. If it is , both you and DM should check entitlements and sync if needed (see link in my signature) DM might also need to turn sharing off and then back on again. 

in terms of compendium only, your DM (who owns Fizban’s) should check their product license.  That’s found on the same page as entitlements (see link in my signature), just further down the page. 

My character is actively part of his campaign. And what is and how do I sync?

Please see the Content Accessibility Troubleshooting FAQ for info on how to sync, as well as other potential fixes to your issue: (https://www.dndbeyond.com/forums/d-d-beyond-general/bugs-support/142100-how-to-access-content-on-d-d-beyond-faq).

Quote from GlareMaster >>

My character is actively part of his campaign. And what is and how do I sync?

 Make sure the DM has Content Share turned on, otherwise nothing they own will show up for you.

Ya, none of that is working. Me and my dm both checked our syncing and he turned sharing off and on again. Still nothing.

If your DM can see this page:
https://www.dndbeyond.com/sources/ftod/character-creation#GemDragonborn

But on the Dragonborn Races page, Gem Dragonborn does not show up:
https://www.dndbeyond.com/races/16-dragonborn#GemDragonborn

Then they bought the compendium content only.

The gem Dragonborn show up for me. I’ll show my dm.

He can see it too. All of it.

Your character sheet might be stuck on an old cache. Try creating a brand new character in that campaign.

When I create new characters it still doesn’t work.

I checked with another person from the campaign and they can’t access the races either. Also, this happens for monsters of the multiverse too. I think it might either be on my dm’s side or it is a bug. But I wanna know if you guys have any idea how to fix it.

I would also like to note I have access to the monsters, spells, items, and feats from shared sourcebooks. I don’t have access to races and classes from them.

At this point, I think you will need to submit a support ticket, (contact us link, below) as we have run through all the usual/known causes and not resolved it. When you do, list all the things you’ve tried already, and specify that your character is in the campaign. 

Ok, I’ve figured it out. Every time i’ve been creating characters or editing my character it was outside of the campaign. I’ve just found it works when editing inside of the campaign. Thanks for helping and trying guys.

How do I close threads?

You don't need to close it, people should hopefully read and realize that is has been resolved. A mod may add flair to mark as resolved.

Ok. Thanks again guys.