I would like to present a newer opinion I’ve been giving a lot of thought to lately. In most situations, I don’t think signature moves should be given to other Pokémon in later generations. Sometimes there may be some cool lore reason for it, like Entei getting Sacred Fire, or maybe it just makes way too much sense for a newer Pokémon not to learn a certain move, like Hitmonlee getting Blaze Kick. But unless otherwise specified, I think signature moves were cooler in the generations where they were stronger moves exclusive to weaker Pokémon.
Giving new moves to older Pokémon is completely fine, but signature moves are often made with their Pokémon in mind- going back to the Sacred Fire example, Ho-Oh loves Sacred Fire’s high burn rate because it has higher Special Defense than physical- this comes with the drawback of breaking the balance of the game if other Pokémon with different traits gain those moves. See Breloom, the first viable Spore user. See Smeargle committing war crimes with Dark Void. See Spikes on Cloyster and later Skarmory. You get the picture. Attack move examples are less common for this kind of thing, I’d say, but the idea that strong moves should be made available for weaker Pokémon still exists, because there is no reason that many Pokémon should be able to learn Close Combat even if it never was a signature move.
Edit: Bulbapedia has it fixed now, but Spikes was never a signature move for Pineco and Forretress. I thought it might have been on Day 1 of GSC but multiple Pokémon got it on level-up.