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I don’t know about all the lion stuff but I’m gonna throw out a guess Rose of Versailles?
"Don't cry because it's over, cry because it happened."
Mmh, no, all the characters seem too old. The girl looked to be about ten.
Definitely sounds like something Osamu Tezuka would do... He had a "star system" where characters of his would play different roles depending on the work, like actors; I wouldn't be surprised if the white lion cub turned out to be a role played by Leo/Kimba the White Lion. If it was an episode of Kimba itself, could the girl traveling with him have been Mary?
Maybe? Kimba has those black spots in his ears and I don't think this cub had them (I'm fairly sure he was explicitly refered as an albino in the show), but I'm not willing to discard it yet. Could you link a picture of Mary? Google Images hasn't been very helpful.
No, that's definitely not her. The girl I'm talking about had yellower hair (Mary's is kind of orange), and it was longer. Around the waist, I think, and kind of wavy.
The art syle was like an old shoujo, everyone's eyes had like three or four dots of light in them, with maybe a few lines too. The colors made it look like it was traditionally animated, which it probably was because I saw it dubbed around 2006. The setting's aesthetics (buildings, clothing, etc) were European. Both of the characters I remember were blue-eyed blondes too, or at least close enough that my memory could distort them into that.
So, the episode starts with a girl, the protagonist, walking into a town. I think this might have been an once-an-episode thing, but I only ever watched one so I'm not sure. I think the girl had a blue dress on (kinda fuzzy there), and she was definitely walking beside her albino pet lion cub. They go near a bridge and see a man standing on the handrails. (They were pretty thick, white handrails, kinda like tiny roman pillars sharing a single top). The girl screams at him not to jump and he turns around, screams and throwns himself onto the bridge's floor while trying to get away from her pet lion. After a few seconds he regains his composure and starts chastising her. He says he's a painter and he was just trying to get a good view of the dawn, but now she's broken his concentration and ruined his work and she should learn not to put her nose where it doesn't belong, etc. Eventually he walks away, partly angry at the girl and partly scared at the lion.
Now, I get the feeling the plot follows the girl around for a while after this, but I can't say much more. The next thing I remember is the artist at a fancy party where nobles (or rich burgueoise, who knows) are chatting about how wonderful his paintings are and such. I think he might have been auctioning them, I feel there were a few shown onscreen. Anyways, he takes a break from socializing to sulk at a corner (might have been a balcony or something, just somewhere where there were fewer guests but still inside the house grounds) and mentally monologues to himself about how he almost jumped to his death earlier that day, but he's got a lion phobia and seeing the cub shocked him out of it. I don't remember if he said it explicitly, but he sounded like he was planning to try again as soon as he could, and he had a sort of "Ugh, that meddling kid ruined it!" kind of attitude.
I don't remember anything after that. I kinda feel like the girl wound up at the party somehow, but that might just be me filling a blank space with my best guess. It was a kid's shows and she was the protagonist, so I'm sure she helped him somehow.