Why does the First Order essentially just make a bigger Death Star?

For a short explanation by JJ Abrams himself:

Starkiller Base was created for the 2015 film Star Wars: Episode VII The Force Awakens. The film's director, J.J. Abrams, stated that the superweapon's similarities to the two Death Stars from the original trilogy were intentional, as the new generation of Imperials within the First Order would want to prove that they were stronger and more advanced than their predecessors in the Galactic Empire.

So basically: they wanted to do things bigger and better than the Galactic Empire did, so rather than construct a space station, they transformed an entire planet and used the acquired knowledge of the now defunct Empire to improve upon their original plans.

And really, that's about the only explanation that's needed. The First Order doesn't just want to be some offshoot vestige of the Empire itself: it wants to surpass them in any way possible to achieve the Empire's original goal. What better way to do that than transform a planet into a super weapon, capable of consuming the entire power of a single star, and jettisoning it out into hyperspace to annihilate any star systems in the galaxy that might oppose you?

The sheer capabilities of this new weapon were far beyond anything the Empire was capable of decades earlier, and shows that the First Order has managed to take what they learned and enhance the Empire's original capabilities exponentially.

They were confident in the weapon because the shielding around the planet had only one weakness: a ship penetrating the shield at lightspeed, only to drop out and right itself at the last second. A weakness for sure, but even Han knew how dangerous it was to plot a course haphazardly when making the jump to light speed in A New Hope. They figured they had eliminated any major vulnerabilities compared to the original Death Stars, and in some ways you can't really blame them for thinking that.

Much of this answer is highly speculative. There are no official sources being quoted.

it was just a ginormous version of the Death Star.

There may have been some practical reasons for this. As noted by MattD's answer, a re-creation that remains may be a powerful statement in favor of the First Order. A re-creation that gets destroyed early on would be particularly humiliating. My guess is that the First Order wasn't preferring to have that happen.

I suppose I could counter with the question, "Why not make it look like a Death Star? Why must the new contraption look notably different?" The question poses some thoughts about this. However, those ideas might even have been considered, but just not be more compelling than reasons for similarities.

It may be useful to define, more precisely, just what is meant by "looking like the Death Star"? Is this observation largely based on the fact that it is spherical?

The fact that this weapon was built on a planet may be one key reason why it had its spherical shape. It could be that the StarKiller was going to be the first part of a much larger lattice/snowflake-like hub. Building around a spherical planet would have led a sphere to be the easiest first step. Additional modifications could be made in subsequent years.

The fact that this weapon needed to contain the energy of a sun may also have had some impact. Since I haven't personally studied the plans of an invention capable of storing that much energy, I don't know just what technical requirements exist.

Maybe a future plan was to ram the StarKiller into the center of a planet, to absorb energy from all directions more quickly.

The Death Star had a sphere that was used to unleash a tremendous amount of power into a focused direction. The StarKiller base performed this same functionality, which may be another reason why there were some physical resemblances.

Many people like their wireless phones to make a "phone-like" sound, despite the fact that the modern smart phones are simply playing a "WAVe" file, which could sound like anything. It could be that the board of directors could have chosen a base that looked like anything, and that similarity to the original Death Star seemed appropriately symbolic of what they intended to accomplish this time.