It's not the best wording, but I believe the shove in Hammering Horns is separate from the Shove detailed elsewhere
Let's look at the Shield Master feat
[...] If you take the Attack action on your turn, you can use a bonus action to try to shove a creature within 5 feet of you with your shield. [...]
Here what it means to "shove" a creature is not explicitly detailed and so it falls back to the general rules on shoving:
Using the Attack action, you can make a special melee attack to shove a creature, either to knock it prone or push it away from you. If you're able to make multiple attacks with the Attack action, this attack replaces one of them.
The target must be no more than one size larger than you and must be within your reach. Instead of making an attack roll, you make a Strength (Athletics) check contested by the target's Strength (Athletics) or Dexterity (Acrobatics) check (the target chooses the ability to use). You succeed automatically if the target is incapacitated. If you succeed, you either knock the target prone or push it 5 feet away from you.
Every aspect of this applies to Shield Master's shove except those that are overruled by the more specific Shield Master feat:
You don't need to use the Attack action, you can knock your target prone or move them, the size restrictions still apply, it is still a check contest, and you still automatically succeed against incapacitated targets.
Let's look at Hammering Horns
Immediately after you hit a creature with a melee attack as part of the Attack action on your turn, you can use a bonus action to attempt to shove that target with your horns. The target must be no more than one size larger than you and within 5 feet of you.
The entire first sentence is nearly identical to Shield Master which means there is a rather compelling argument that this also refers to the standard shove detailed above.
Note that Hammering Horns then explains the size restriction which would've have already been covered by the typical shove requirements. This isn't really evidence of anything though since this wouldn't be the first instance of the technically unnecessary reiteration of rules.
Hammering Horns then mentions new mechanics
Unless it succeeds on a Strength saving throw against a DC equal to 8 + your proficiency bonus + your Strength modifier, you push it up to 10 feet away from you.
If this were about the traditional shove, then both the typical check contest and this saving throw would occur; this would all happen in addition to the effects of a regular shove. This description mentions nothing about replacing the check contest, only that when using this shove a saving throw also occurs. This means there are two possible interpretations:
Hammering Horns either replaces the typical shove or happens in addition to it
If Ramming Horns was done in addition to the pre-existing shove mechanic, it would have to be exactly that, an addition and not a replacement. This means that you would have two options: push the target 15 feet away, or push them 10 feet away and knock them prone (or mixing depending on how the contest and save go).
And if Ramming Horns was a replacement for the pre-existing shove mechanic, it would mean your only option when using Ramming Horns is to shove your target 10 feet away.
I believe the latter of the two is the case despite the similar wording to Shield Master. I can't really give a good rules reason why but if anything else were the case I would have expected the feature to say something along the lines of "In addition to the typical effects of a shove..." or "If you choose to shove your target 5 feet with this action, you instead shove them 10 feet".