I was recently gifted a female dragon and a female sygmurh but the breeding option is greyed out on both. Is this a bug, or can we not breed gifted pets?

My guess is that your stable is full. I count 19 pets and one of your Dragons is pregnant. The baby already counts towards your stable, which means you are probably at 20/20. You will need to get a Blueprint: Stable (lvl 2), then get the materials and build it from the Craft & Construction menu. 

The blueprint can be found in Explore or you can buy it from the Item Trades or Auctions. There is also the option to buy it from the State Store, but I wouldn't recommend that. Their prices on blueprints are seriously overpriced ;)