I believe it is his demon form.
Remember when you die in SotN, the picture shows a demon skeleton with a crucifix.
I always assumed that the "gargoyle" was a manifestation of his demon form that is brought out by Alucard's anger at being petrified.
As for the game over screen, I always associated it as a reference to Dracul, the dragon.
Guys, guys, there is nothing profound or symbolic about the Gargoyle form. Depending on the brilliance of the director/producer, little details with subtle meanings can be quite common in movies and shows ("Breaking Bad" and the first "Matrix" film are great examples) but in video game production the art direction seldom has this much influence, even when it is one of the biggest selling points, as is the case with the CV series. Besides, symbolism and deep developed characters aren't IGA's style.
Crisis pretty much answered his own question:
ive wondered over the years, could this in fact be alucard's "demon" form emerging? being the son of the king of the underworld, why wouldnt he have a demon form like his father? whether the developers intended this, or just them being silly by adding an odd easteregg, i thought it was creepy & exactly how Alucard would look if he were to transform. i recall he even sprouts wings during one of his finisher techniques in Judgment
At best, it was a leftover sprite for an unused form concept that a low-level programmer added in there for shits & giggles; probably the same guy who felt that pizza was an appropriate food item for a game set in the late 1700s.
There were so many things in SOTN that were untapped and random. Given the size of the game (at the time) the amount of items one could find, the amount of familiars to level up (more than enough for 2-3 playthroughs)
Something that happened randomly a few times, I found that if I stood at the very top of a spire in the Holy Chapel with the Sword Familiar active, the music would stop and it would say something for about 2 seconds, not when attacking, not when doing Sword brothers, just being stationery. I can remember the exact phrase, but it was creepy. I think coupled with the Gargoyle and the bleeding fountain, these crept me out the most. (Personally I felt as if Castlevania stopped being eerie after AoS)
The demon & fairy familiars also have something to say if you remain stationary for a few moments after summoning them, regardless of the location. One of the coolest things about SotN is that it is just full of little details like that. Ever approach the skelerangs? They'll cower in fear when you get close to them. The "Verboten" banner that the Rune Sword makes is a nice touch too, though I've personally only ever seen it in screen shots.
The music on the second floor of the Theater in LoI was eerie as all hell, as I felt the same about the "Nightmare" track in CV3 as a kid. The Villa in CV64/LoI wins though for eerie atmosphere and horror,