The most critical component of a Metroidvania (which these days really just means "Metroid-like") is that the player's progression through the game world is gated by their acquisition of new character abilities and upgrades, not just keys. In this sense, Dark Souls is very obviously not a Metroidvania. The player's mechanics remain constant through the game other than their choice of weapon moveset and potentially some utility spells or equipment perks which do not affect progression, with some very minor potential exceptions. The only difference between the cursed undead you start as and the one you end the game on is their stats and gear, which are mostly relevant only to combat. However, this does not mean there aren't some elements of Metroidvania design in Dark Souls. Ultimately, the progression gates in Metroid-likes are just locks with a different type of key, namely the ability to double jump or go into morph ball or what have you. Hence, while Dark Souls will never provide those same eureka moments of "aha, now I can jump up to that one place!", it does have a similar design around backtracking and making note of locations so you can return to them later once you've acquired the correct key item.
Moreover, Dark Souls is very spiritually in tune with the -vania part of the Metroidvania moniker, to a much greater degree than most Metroid clones on the market. Exploring the game world and uncovering secrets, fighting a large bestiary of enemies and tough bosses, leveling up, finding new gear, and developing a playstyle based on a wide suite of combat options all feels very similar to the latter-day Castlevania games (Igavanias). Obviously the pacing and tone of the games is very different, and Dark Souls introduces some new twists with its death and character upgrade systems, but Dark Souls feels very analogous to Symphony of the Night as the methodical 3D counterpart to its brand of 2D sidescrolling hack 'n' slash jRPG action, in part because it's coming from the same rich design tradition (Dark Souls did not invent this stuff, people). In this vein, Dark Souls would deserve the Metroidvania name as much as a pure Metroid clone like Axiom Verge if the latter part of the genre name were to signify anything at all. But, I know it doesn't in its popular usage, so Dark Souls isn't really a Metroidvania as such.