Anapanasati is a Pali word, and as such, it has its own complete practice in Theravada that covers both samatha and vipassana. In East Asia mindfulness of breath is mostly used only for samatha practice, and as such, it is only for calming the mind.
Zazen means sitting meditation, literally (za - sit; zen - meditate). Meditating in the sitting position can mean any kind of mental training. You can count your breath, just watch it, or be mindful of your body, your thoughts, visualise a buddha, etc.
What people do within the Japanese Rinzai school depends on the lineage and the teacher's instructions. Breath practice is common, sure. It's a kind of preliminary to koan practice. But shikantaza is a different approach, and it's not the same as just objectless meditation. Of course, we can say that ultimately koan practice leads to shikantaza. Or we can also say that shikantaza is mistaken and koan practice is correct. It depends on who you ask.
"While teachers of the middle way, mind only, transcendent wisdom, mantra, and other schools may have their own assertions, the fulfillment of those intentions is the same. There is not a single thing that is not contained within mind."
(Gampopa to Düsum Khyenpa, in "The First Karmapa", KTD Pub, p254)
“If you recognize the world of appearance and existence as the mind, realize the mind itself as empty, and have no grasping at the superiority of your realizations — this is the ultimate view."
(Chegom Dzongpa, in "The Book of Kadam", Wisdom Pub, p609)