Of course, a Tempest Cleric will already be hurling lightning spells, so a wand of lightning bolts doesn't seem like it should be outright "no clerics allowed."
We keep defaulting to the old the arcane/divine divide. Unless I'm wrong, that divide ONLY appears in 5e in that one sidebar in the PHB magic chapter talking about the Weave (a.k.a giving a mechanics-derived fluff explanation, based in previous editions' history, for D&D magic). Otherwise, classes that use magic in D&D 5e use "Spellcasting" – they have separate lists of spells known per their different means of learning/preparing spells by class (and thus their method of learning magic), but share spell slots even when multiclassed. The only distinction in the Pact Magic of Warlocks, which is specifically a cheat given by eternal powers (compare it to clerics and the gods like comparing teachers tutoring a student before the big test versus one sneaking the answer key to the same student when no one's looking...)
Within the mechanics, needing wands or such to require Spellcasting or to have the specific spell(s) on the user's [class] spell list is a reasonable limit now that you can't hardwire the same "this is divine" and "this is arcane" in quite the same fashion. For more common spells that are present on more lists, you will see more crossover, but that's also a benefit for choosing a specific subclass (e.g. the Tempest Cleric of whom I spoke above, plus his Light Cleric pal, gets to use more of the good combat wands than traditionally expected); those certain cases are not indicative that every character everywhere will be running around looking like a tongue dispenser case.