Two Quintillion, give or take
Assuming all we care about is what you get directly from being a Fighter. The Fighter makes four categories of decision; let's cover them one by one.
As you say, Fighting Style is chosen at level 1, which gives 5 options.
At first level, fighters choose two skills from a list of eight, which gives 28 options total.
First there's the Champion. At 10th level, Champions get a second Fighting Style; this feels like it should be four options, but since we don't care what order we get our Styles in there are only ten possible pairs, so the multiplier for this is 2.
Next up is the Battle Master. They choose five manoeuvres by level 12, from a list of 16; that's 4368 possible choices. They also get proficiency in one of 17 sets of artisan's tools, for a total of 74256 Battle Masters.
Finally, the Eldritch Knight. By level 12 they know 8 spells; they can be any combination of 1st and 2nd level Wizard spells, and 6 of them must be Abjuration or Evocation. 15 spells exist that satisfy all these conditions, and 49 satisfy all but the School requirement. So they can have (15C6 * 49C2) + (15C7 * 49) + (15C8) = 5005*1176 + 6435*49 + 6435 = 6207630 spell choices. Add to that 3 of 16 cantrips, a total of 560 cantrip choices, and you get 3476272800 possible Eldritch Knights.
Sum the classes and you get 3476347058.
A 12th-level Fighter has 4 ASIs; if all are spent as ASIs, then the player has 8 points to distribute over 6 skills, which is 13C5 = 1287. If 1, 2 or 3 are spent as ASIs, then the total number of choices is 21, 126 or 462. Great.
The Player's Handbook offers 42 feats; they are also incredibly complicated. Perhaps later I will edit in an analysis of what the individual feats do to the maths (spoiler: it's hecking complicated), but for now I'm just going to pretend that each feat can be taken once and requires no decisions. So for 1, 2, 3 or 4 feats, you have 42, 861, 11480 or 111930 choices. Cross-multiply these numbers, and you find yourself with 2350530 + 1446480 + 397782 + 54054 = 4248846 choices here.
Put these options together and you get: 5 * 28 * 3476347058 * 4248846 = 2.0678649e+18. That's just a hair over two quintillion options.
This is a bad way to prove your point
You state in the comments that "This would be a useful point for subsequent "it's all the same" and "feat/ability trap" discussions". This is wrong. It would not. Two quintillion is a nonsense number. It enumerates such helpful character choices as follow:
- A fighter with 14 Charisma and 15 Wisdom, and the same character with 15 Charisma and 14 Wisdom
- A fighter who took the Resilient (Strength) feat for +1 Strength and no other effect, a Fighter who took the Weapon Master feat for +1 Strength and no other effect, and a fighter who took the Heavily Armored feat for +1 Strength and no other effect
- 1364 Battle Masters who only use Precision Attack and ignore their other four manoeuvres
- An Eldritch Knight who can cast Light, and one who can cast Dancing Lights
- A Champion who put all 8 points of ASI into Intelligence
And each of these characters overlap and are repeated millions of time, along with many others. I have not done anything helpful here; I've done a fun maths problem, and this answer does not deserve a place in any halfway serious conversation.