This is an interesting situation. Although you can use the 'normal' checks as listed above, I would take a slightly different approach (although still able to be adjudicated through the normal rules). Sometimes the basic attributes for a skill don't cover all its uses.
I would base the skill proficiency on the type of animal you are trying to emulate. If you are trying to emulate a domesticated animal that is possibly trained in tricks, Animal Handling seems the most appropriate skill to use. If you are trying to emulate an animal that is wild and trying to do so without appearing to have more intellect than it does then Nature may be a better option. If you are trying to emulate something truly extraordinary, such as an elemental or other magical creature (polymorph), then Arcana is likely the best skill option. Choose the skill that is most likely used to identify that creature and its actions and you should be on the right track.
Fooling The Enemy
Any time you are trying to trick your enemy into thinking what you want, you need Charisma, so this choice is obvious. Yes, it is a form of deception, so that skill may be appropriate in some ways, but I would say a druid is likely to be far better served by knowing how the animal acts and acting based on that.
Due to this my suggestion is a Charisma check with proficiency modifier based on the "Which Skill?" decision above.
When observing your surroundings, Wisdom is the go to stat, so this is the right stat to look at here (unless suspicion is aroused, then Intelligence may be a better option such as an investigation check would lend). Accompany this with an appropriate skill as noted previously and you should be on the right track.
Wisdom check with proficiency modifier based on appropriate skill.
Other possible methods:
Another possibility is giving advantage to 'normal' skill checks if they are proficient in a related skill, but this seems a bit over the top to me. You end up giving out some form of advantage/disadvantage on every single roll.
Finally, you can just use standard skill checks as they are normally rolled. I find it a bit odd to use an Intelligence(Nature) check to determine if someone is an animal, but a Wisdom(Insight) check doesn't quite seem appropriate. Charisma(Deception) seems wildly inappropriate for a druid who wants to act like a creature they're likely very intimate with the actions of in the first place, so I lean towards my above suggestions.
Wrap it up (TL:DR)
Summation (uses a horse or dog in examples):
Choose a skill appropriate for knowledge about the creature and its
Shape-Shifter rolls Charisma check, with proficiency as noted above. This is the DC of the observer's check.
- (Charisma(Animal Handling) check)
Observer gets a passive check if not actively searching.
- (10 + Wisdom modifier + (Animal Handling))
Observer rolls actively if trying to detect abnormalities.
- (Wisdom(Animal Handling) check)
This is only to cover how to deal with the roll. My general adjudication would say that if they are significantly more proficient, and have advantage (due to role-playing for example), then there is no need for a roll.
I have no hard and fast rule on that, but an obvious scenario would be the trained and specialized druid (+3 Charisma, +2 prof bonus) with advantage (+5 to passives) would have an "average roll" of 20. That is a no-brainer in situations where the bonus and advantage give 10 points more for the shape-shifter than the observer has. A close check is a bit different, however. An average guard would have an average roll of 11-13. This means that they would need about a 17 or higher to notice the difference. Let them default the effect with no roll.
It's not really that hard, especially in 5th edition, to keep track of these numbers. In this case it's Charisma mod, proficiency, and advantage/disadvantage. If all are positive, lean towards defaulting the roll to success.