Hi Gina! No, it's not an adoption record, it's the birth record of a child that was left at the ruota dei Proietti, a foundling wheel widely used in all Europe to give up babies for adoption without revealing the identity of the parents.
The record is signed by the Sindaco, probably because he had the authority to give the foundling a name - he's not adopting it. Tha baby was male and appeared to be 2 days old, no remarkable physical traits. He was named Carlo di Giuseppe (every region/town had its own traditions to name the foundlings).
Regarding the link to this birth records book, it's not indexed yet, so the individual pages do not have a link each, but only one link for the whole book. If you're going to cite it, you need to add the number of the page after the rest. You can also use the very useful Greg Clarke's Antenati Citation Builder that you can find here: https://apps.wikitree.com/apps/clarke11007/antenati.php.