costumed adventurer: always in costume as "those who look upon Phantom's face must die". Even when the Phantom goes abroad as 'Mr Walker' (for 'the Ghost Who Walks') he just basically throws a hat and coat on. How this works for the passport photo is one of those great unanswered questions.
So, yes, while it has been said that Bruce Wayne is a guise that Batman employs, rather than vice versa, and Clark Kent is Superman with a pair of glasses and his hair combed differently (which doesn't work as a disguise, in case you were thinking of trying it), The Phantom has no other identity at all.
skull cave: the batcave and arrowcave are situated under their respective heroes' mansions to allow for easy access but the skull cave is deep within the jungle so good thing the 'Guardian of the Eastern Dark' has a horse. The Fortress of Solitude may be way off in the Arctic but that's not a problem when you've got superspeed.
secret command: the Jungle Patrol have a secret commander who sends and receives messages via a well. He also received messages by jungle drum. This may have worked in the nineteen thirties but now he'd communicate via fax or e-mail. Not half as romantic.
pets: for archetypal names you can't go past Hero and Devil. And the horse and wolf make sense as offsiders. Ace the Bathound always seemed ridiculous and Krypto, Comet and Streaky even moreso (what's the point of a flying horse when you can fly yourself?! - at least the wing-ed steed is useful to the Shining Knight and the Black Knight)
love interest: while supes occasionally fraternises with other LL-monikered members of the opposite sex, the majority of Marvel heroes have been serial monogamists, and Green Arrow sows his seed with wild abandon, this generation's Phantom has been a one-woman man.
realism: no supertype baddies, only pirates, corrupt sultans and common crooks. The Phantom is well developed, lightning fast etc but capable of being injured.
weapons: like all the early heroes, uses guns and his fists. Most of the lineup of the Seven Soldiers of Victory were characters who got dressed up in gaudy outfits and punched you in the head. That was it (so theoretically anyone with a propensity for violence and taking the law into their own hands could have become a superhero.)
successors: The Phantom has the most prevailing need to have an heir. For otherwise the legend ends. There are other heroes who pass the flame to their own progeny vizStarman and The Ray.
But most often they are wards or junior partners - Kid Flash becomes The Flash when Barry Allen dies (he, in turn, was inspired by Jay Garrick, the Golden Age Flash); Robin the Boy Wonder.
spectre of immortality: quite common in the pulp era heroes. Denny Colt survives death to assume the mantle of The Spirit. Another detective, Jim Corrigan, is shot to death and is given a chance to exact revenge in the all-powerful figure of The Spectre.