Do they have to be the same type of species?

Yes, you can only breed the same species together.

You can breed your own two creatures together at the Breeding Grounds, or if you'd like to breed one of your creatures with someone else you can do so through the Advanced Search.

1. they do have to be the same species, there is no hybridization here
2. both pets have to be Adult to breed
3. the female cannot be on breeding cooldown
4. a pet cannot breed if someone has made it infertile via infertility kit.

2 elaboration:
To check if a pet is Adult, you can go to their page and check the Maturity stat.
If a pet has been genetically tested or was made in the generator (such as starters), you can see their exact Maturity Hours, if not, you can only see Eggling/Hatchling/Adolescent/Adult

Example, your Qwant is, at the time of me typing this, "Maturity: Adolescent (219/648)"
She's a starter, and thus counts as a Generated pet, so shows her Maturity Hours.
Generated pets start at the beginning of Adolescence, 216 maturity hours
A pet reaches Adulthood when they reach 648/648mh

a pet gains 1mh per hour, but mh number can be increased via Nurturing on egglings and hatchlings, and via magical pie on adolescents.

(more in-depth breakdown on aging in this guide post 5. Guide also contains other handy info such as how sylesti genetics works)

3 elaboration:
if a female is on cooldown, she will have this symbol in the stables, and will have text at the bottom of their pet page that says "*Mated with [x]* Recovering Until: [date]"

4 elaboration:
Infertility kit usage cannot be undone.
if a pet has been made infertile it will have this icon on their pet page above their genes