Rules as written, an enemy is anything your DM allows you to consider an enemy.
What is an Enemy?
Enemy is not clearly defined, so there is no clear definition of what is and is not an enemy. It is up to your DM to decide. If you want to give justice to the intent of the rules, then I would say an enemy can only be a creature. Throughout the PHB "enemy" is used only as a descriptor of a creature when we're looking at the rules of a given mechanic or ability.
Some examples of this:
Warding Flare (PHB 51) Also at 1st level, you can interpose divine light between yourself and an attacking enemy. When you are attacked
by a creature within 30 feet of you
Holy Nimbus (PHB 86) Whenever an enemy creature starts its turn in the bright light, the creature takes 10 radiant damage.
Elder Champion (PHB 87) Enemy creatures within 10 feet of you have disadvantage on saving throws against your paladin spells and
Channel Divinity options.
And finally, a non-mechanical use of enemy. Flavor text uses enemy in the way you'd expect.
Ranger (PHB 89) Clutching a shortsword in each hand, he becomes a whirlwind of steel, cutting down one enemy after another
When is a creature an Enemy?
A creature is considered an enemy when you consider it one. And, you can only do so if your DM allows you to. More on that later. Enemies and allies are defined by one creature towards another. A creature cannot define its relationship from the perspective of another creature. I cannot define that Charlie considers me an enemy, because I'm not Charlie. Charlie might well agree with me that we are enemies, but Charlie is the one defining how he feels towards me. In that sense, any creature can be an enemy.
There are even instances where a creature is not aware that you consider it an enemy. Would that mean you can't treat it as an enemy for abilities? Absolutely not, that would destroy any predication for stealth and deceit, since you wouldn't be able to cast spells or abilities that target enemies without first making that known to them.
Furthermore, enemy is used in the rules to differentiate who the targets of harmful spells are. "Friendly" spells are never predicated by only being able to target friendly creatures. They can target all creatures. The phrase enemy creature is used in spells so that your actual friends can, for instance, enter your aura of menace without being frightened.
By contrast, a fireball doesn't hit "enemies" it hits "creatures", meaning everyone. When enemies are singled out, it is only used to differentiate that your allies and non-enemies are not affected. As such, based on the rules as intended, I wouldn't allow you to use Aggressive on friendly or non-hostile creatures, as the intent is for you to get closer to your enemies so you can hurt them, not for you to run away or move quicker towards an objective.
Your DM will define what it means to consider a creature an enemy, since the rules do not. Your DM may say
"You may use Aggressive on that NPC, but he will become hostile to you because of your body language and expression."
He may even ask you to justify it, and could even shoot down your attempts.
"You can say your Orc sees this squirrel as an enemy, but I don't believe he can flip a switch that easily."
The onus is on your DM to keep this ability in check for you. Can your character really, honestly say a tree is his enemy? Even if you say "Me find tree insult my race, me hate tree, me ANGRY!" is it actually true? This takes us into the the realm of defining your character and what your DM deems to be reasonable. I would shy away from defining a character on the basis of mechanics, although it's not against the rules. If you've a convincing backstory as to why you hate trees, then by all means a tree can be your enemy.