The pet still gets its ability improvement, even if its master takes a feat; the language here is simply informing you at what level the pet gets this feature, it does not get "skipped" depending on the character progression choices of its owner. Whether or not a pet could take a feat should be left up to the DM to decide.
From Twitter:
YDMND - UA revised Ranger: if the ranger takes a feat insted of ASI, does the companion still get ASI?
Mike Mearls - I'd say yes
Balancing Your Feat
I don't think it's terribly well balanced; in most cases, it's actually weaker than a feat:
If you take the first option twice, it's the same as a basic ASI.
If you take either the second or third option, it is strictly weaker
than Skilled (PHB p170), which allows choice of three skill or tool
proficiencies.
Take the first and last option, and it's weaker than Linguist (PHB
p167), which provides +1 Intelligence, three languages, and the ability to
create coded messages.
Feats are supposed to be powerful, and your design is watered down in comparison. Feats also tend to be packaged abilities with a theme, selecting one makes a character different from others with the same class. Were I playing in your game, I wouldn't take anything but the ASI, which I could do anyway.
There are also stacking concerns - normally, the same feat cannot be taken more than once. If this is the only feat available in your game, then it's clearly stackable. It allows a player (over time) to stack a large number of skill proficiencies, potentially having proficiency in everything, which will trivialize skill checks and reduce each character's uniqueness.
The Power of (Variant) Humanity
Variant human is pretty powerful, but so is a normal human. I think you're overlooking the raw strength of solid attributes. Having a bonus to six stats shores up potential weaknesses and/or saves points the player can spend on the character's desired strengths.
New Player Advice
If you're trying to ease new players into the system, you could disallow "Variant Human" completely. That would push off selection of feats until L4 at the earliest (if you allow them at all, they're entirely optional), and allow some time for people to get used to the basics.
Best Answer
No, you cannot choose to get a +2 ability improvement instead of a feat
No rule exists allowing a feat to be replaced with an ability score improvement. Instead, the opposite is true:
Page 165 PHB, Feats
Thus, the class feature Ability Score Improvement can be replaced with a feat and feats cannot be traded for ability score improvements. It should also be noted that this rule specifies that the trade can only be done with the specific feature provided by your class only, making ability score improvements from racial traits ineligible for trade.