Does the racial feat Fade Away count as a spell cast by the user or is it a racial trait activation such as a dragon's fire breath and therefore can't be counterspelled?

No, Fade Away is not a spell

As Jeremy Crawford, lead rules developer for D&D, has said here:

Want to know what spells look like? See "Spells" (PH, 211–89). Nothing else is a spell unless it is presented as a spell or called one.

Fade away is a racial feat that allows you to do certain effects including:

Immediately after you take damage, you can use a reaction to magically become invisible until the end of your next turn

However, nowhere does the feat saying anything about casting any spell. Without something telling you that you are casting a spell, you are not casting a spell. This is simply a racial feat giving magic abilities.

See this Q&A for more on what is and is not a spell.

Counter spell only works on spells:

You attempt to interrupt a creature in the process of casting a
spell.

Since counterspell only works with spells, it will not work with Fade Away.

No Counterspell

The text for the Racial Feat Fade Away states (emphasis mine):

...you can use a reaction to magically become invisible until the end of your next turn...

As you are not casting Invisibility, but simply becoming Invisible, there is nothing to counterspell.

However, someone could still cast Dispel Magic on you as you are under a magical effect and dispel the invisibility.