There seems to be a lot of confusion with this achievement when in reality there is nothing to it. It's all about knowing which animals are missable and/or take extra steps to unlock. It took me a few playthroughs to make sure these were in fact all the pets.
The first missable pet is of course Major Tom found on a small broken structure in the beginning ambush site.
The second is Billy and Jean which you get from saving the Hoons at Hoon Homestead after you venture out from Colorado Springs for the first time. You must save them to be able to tame this pet.
The last that requires special steps is the Bison. They are found at the Knox Bison Ranch which is located between Gett Family Homestead and the Abandoned Oil Well. In order to unlock this location you both have to finish the quest with Victory Buchanan(arrest/kill/squad), and accept the quest to locate the runaways from Cannibal Jamboree. Once at the ranch you have to defeat the scar collectors attacking the ranch. Make sure you have animal whisperer 9 and it will show up in the dialog options after the fight.
The rest are available throughout the map as listed in the other guide. I'll repost here to make things a little easier and add a couple extra locations I've found. You can have an animal for each member of your squad. The only way I found to do this way click the button(R3 or right joystick) that let's you control only one character and then use that character to charm the animal.
Fair warning for those doing the book drop glitch, if the first point towards animal whisperer is from a book, then that character will not be able to tame any animals even if you get them up to 10 with the book or use skill points after.
Dog - hq - colorado springs
Cat - hq - colorado springs
Chicken - garden of gods
Fox - garden of gods
Rabbit - garden of gods
Mountain goat - gett family homestead and denver(2 in the first location off to the right as soon as you enter Denver)
Waste wolf - bizarre
parrot - bizarre (InaneVelen)Stated in the comments that he never purchased Polly and still received the achievement.
Pig - warrens
Stag - Yuma county at godfishers camp, Aspen before taking the Gondola up, and outside the airport of Denver Airport.
Doe - Yuma county at godfishers camp, Aspen before taking the Gondola up, and outside the airport of Denver Airpor.
Honey badger - massacre site nw of Yuma County. Animal whisperer 10 required.
Growler cub - aspen
Razorback - can be found at Department of Energy Site - ( synth head hunter quest) and when returning from the machine commune there is a Razorback being attacked by waste wolves.
Hopefully this helps those still working towards this achievement. Let me know if you're still having trouble and I'll try to help. Thanks and happy hunting.