There seems to be some confusion around what, specifically, you need to get pictures of to meet the requirements for this achievement, so I have noted exactly what I took pictures of. The list is as follows:
HIRSCHFELDEN
Roe Deer - Female
Fallow Deer - Male (story mission)
Red Deer - Male
Red Fox - Male
Wild Boar - Female
European Bison - Female
Canada Goose - N/A
European Rabbit - N/A
LAYTON LAKES
Blacktail Deer - Male
Whitetail Deer - Female
Coyote - Female
Black Bear - Male (story mission)
Moose - Male
Roosevelt Elk - Female
Mallard - Male
Jackrabbit - Male
Mission photos count as I did not take photos of either the Fallow Deer or black bear outside of these missions, and it does not seem to matter whether the animal in the photo is male or female in any case.
The game should, indeed, display the autosave icon at the bottom of the screen once you have successfully captured a photo of an animal for the first time. As stated above, the achievement popped for me immediately after taking a photo of a male jackrabbit.
In my experience, you typically need to be within about 50 meters of an animal for a photo of them to count (again, this was approximated based on the appearance of the autosave icon.) It can be tricky to get close enough to some animals without scaring them so, if you feel so inclined, you can use the following glitch to decrease your visibility until logging out of the game. It is certainly useful for some of the more elusive species like red fox and coyote.
It should be noted that I was not the one to discover this method and all credit should go to the individual that made the video.