The first encounter with an iguana in Cyberpunk 2077 takes place if the Nomad path is chosen - the big lizard is supposed to be smuggled into the city. The second opportunity to see an iguana is available regardless of origin - you will see it during The Heist mission. This begs the question whether an iguana can be adopted. You will find the answer on this guide page.
Is it possible to adopt an iguana?
Unfortunately, iguanas cannot be adopted. Only one small interaction with the lizard is available, but there is no way to adopt it permanently and e.g. move it to V's apartment. If you chose the Nomad prologue, you will have a chance to see the iguana very quickly - you will find it in the smuggled crate. This is not the only opportunity, because the iguana is then located in the Konpeki Plaza, which you will visit in one of the main questline missions.
However, this does not mean that you can't have any pets in the game. In Cyberpunk 2077 you can adopt a cat. On top of that, it is actually very easy to do. You simply need to find a bowl on the first floor of the megabuilding where V's apartment is and put the cat food there. The exact way to get the cat is described on a separate page of the guide: The cat - how to adopt it?