You don't need to "register" a POA GENERALLY, as you would "register" or "report" a live birth, a marriage, your DD-214, etc...
You only need complete the form and have a Notary Public acknowledge and record those who sign the document.
There is ONE exception to "reporting" or "filing" a POA in NY state.
If the POA is for real estate transactions, it must be "filed".
The Power of Attorney if it is to be used in a real estate transaction must be "filed" with your County Clerk's office.
Be further advised, that after you have filed the POA with the County Clerk's office, your Power of Attorney becomes a public record open to inspection by the public.
If the POA is ever to be revoked, a writing to revoke the filed Power of Attorney should also be filed in the same County Clerk's office.