There are ~8 pets that are shared across the four Covenants, obtained in different ways depending on which Covenant you're in.
This overview explains how to obtain these battle pets as a member of the Night Fae.
Many thanks to @patf0rd for their help testing.
If there's anything I've missed or is incorrect, please let me know, thanks!
Covenant (Sanctum) Feature
The Covenant (or Sanctum) Feature that yields these pets is called The Queen's Conservatory. Think of it as a "talent" for your hall/sanctum, and it can be upgraded multiple times. Each Tier upgrade unlocks additional content and provides bonuses.
Once the feature is unlocked, you'll gain access to what I like to think of as a private garden atop one of the gigantic Ardenweald trees. In addition to growing plants, your ultimate goal is to help the dead (Spirits) grow enough to be able to transition back into the realm of the living (reincarnation).
You'll start out with a modest number of pods and plots to work with. As you apply upgrades more pods and plots will become available.
Each Tier upgrade costs an increasing amount of Reservoir Anima and Redeemed Souls, and will take an increasing amount of time to complete.
For the pets, you'll want The Queen's Conservatory, Tier 2 (Initial Growth) at the very least. This unlocks at least one Wildseed pod that is attached to two Catalyst plots. More is better, of course! Additional information on why this is the case listed below.
TL;DR -
1. Upgrade Conservatory to Tier 2. (if you use your Spirits prior to this is up to you, but not worth it IMO)
2. Get 3 Wildseed Root Grain.
3. Place the 3 Wildseed Root Grain in dirt piles so that all three are touching 2 pods. (see image below, where the #4 is the area you're targeting, so those 3 connecting dirt piles + 2 pods)
4. Make sure the 2 pods have Wildseed Root Grain buffs each. (can click on them to check)
5. Insert the Spirit of your choice into each pod. (each category of Spirit has 2 possible pets each, Prideful = Venthyr-themed pets for example)
6. Wait for however many days for your Wildseed to mature. (can click on the pod to check its buff for time left)
7. Open the mature Wildseed, talk to the Spirit who will spawn a chest. Pet drops from the chest and NOT the Artisan Satchel that also appears in the chest.
8. Repeat until you get all the pets! Good Luck!
(see
this list of which Spirits drop which pets)