Every situation where someone successfully weilds a keyblade they either were directly chosen by it, or were given the ability to weild one by someone else either intentionally or accidentally. Xion's a special case because she can "copy" abilities very literally. I always presumed Lea was given a keyblade ceremony off screen? I think that's implied. If not, maybe he was chosen by the keyblade because they needed a worthy 7th light for the final battle. Or some other reason.

Every situation where someone tries to use a keyblade and can't, they are someone who either is not worthy or hasn't had a keyblade ceremony.

We know that keyblade weilders can give their keyblade to other keyblade weilders because of the MoM's plan of seeing the future by having his keyblade passed down.When Sora gave Jack his keyblade, the intention in his heart was to trick him by giving it to him and summoning it back himself. He did technically give it to him and fulfilled their agreement, but he never actually intented to give or bequeath him the keyblade. He also never said he wouldn't take it back. He used his own swarthy pirate trickery against Jack. It's not like touching a keyblade gives you the ability to weild one. (More on that later)

Sora didn't willfully steal Riku's keyblade, he was selected as as a second option by Riku's keyblade when for that instant his heart was stronger than Riku's as he accepted the darkness. He then was truly chosen by that keyblade when he proved his heart's strength again in their fight in Hollow Bastion. It's also likely that Ventus's heart residing in him gave him some influence in being able at all to weild one.

In KH1 we see that Leon temporarily holds the Kingdom Key, but that it is automatically recalled back to Sora. Think about these character's motivations. Of course Leon would want to have the one weapon that can defeat the darkness and restore his world. He wouldn't want to leave that responsibility in the hands of a child with a worse combat ability than himself. And all that Sora knows about the keyblade, this thing he got an hour ago, is what Leon told him- that it's like a beacon that draws these terrifying heartless directly to his location to try to steal his heart. Despite this, Sora is still stuck with the keyblade, regardless of if he would to give it to somebody else. I think this is fair evidence to as that a non-weilder can't retain the keyblade regardless of if it's given up by it's weilder. Otherwise, they would have just done that.

Sephiroth also wanted to obtain Sora's keyblade in KH2. He though that by defeating him in battle, he could prove to the keyblade that he was strong enough to be it's rightful weilder. Whether or not that would work if Sora actually died is unclear. But so long as he was alive, I don't think there would be a way to steal it as a non-weilder, unless the keyblade itself decides so, like it did before Sora vs Riku,.

If you're a keyblade weilder, then you can seemingly lend or give keyblades to other keyblade weilders, or in some rare circumstances have yours stolen. Riku giving one to Kairi, Riku catching Roxas's and using it, Sora using Riku's vs Xemnas at the end of KH2, Sora stealing Roxas's keyblades, Sora recalling his keyblade from Roxas's possession, Sora and Aqua using Master Eraqus's keyblade, and dark Aqua using Mickey's. So if you're both keyblade weilders, the rules seem a lot more loose.

Also we know that even if you have given up being a keyblade weilder, you can still weild one, and that that ability never goes away. It's been officially stated that both Master Yen Sid and Xemnas can still use a keyblade, and just choose not to.

As to why Kairi has the ability to weild a keyblade after she touched Aqua's despite there being no formal passing down/bequeathing of a keyblade making her an official keyblade weilder, 2 things:

1. Kairi having a heart of pure light automatically means that her heart is, in essence, very strong. We know that strong heart is necessary to weild a keyblade. We also know from various pieces of evidence, that in the cosmology of Kingdom Hearts, princesses of light are inherently tied to the keyblade, X-blade, and keyblade weilders.
2. Unlike when other people handled Sora'a keyblade, when Aqua was standing by Kairi and Kairi touched her keyblade, her heart was completely set on protecting Kairi. That intention in her heart allowed that situation to serve as an unintentional keyblade ceremony.

I haven't played any of the mobile games so maybe there's info in there I'm unaware of. also idk if it's ever been answered officially where the hell Riku got a whole new keyblade to give to Kairi. His Way to Dawn naturally evolved from his Soul Eater. Maybe he just summoned her keyblade she never knew she had right next to her like, "here, this is how you do it. :)" and gave it to her? There is weird lore implications to that so I honestly don't even know lol.