They almost certainly would have, but sources are limited and a full list of the dozens of ships in the fleet is, to my knowledge, not available.

For names more generally, the Qián Wén Jì (前聞記, which roughly means an historical account, but poetic) by Zhù Yǔnmíng (祝允明) offers some typical ship names of the period, but not identified specifically as belonging to the treasure fleets. The names given are 清和 Qīnghé, 惠康 Huìkāng, 長寧 Chǎngníng, 安濟 Ānjì and 清遠 Qīngyuǎn. These could translate as Pure Harmony, Bestowing Peace, Eternal Tranquility and I guess I'd go with Clear Distance on the last one, which works for a ship. Zhù adds that most of the names were of this sort. There's a clear theme here, and like you said, pretty straightforward. I'm not sure what your Google search brought up, so if you'd share I'd be interested. I imagine it's probably coming from the QWJ.

A common source for Zheng He specifically is the Ming History (明史) which has a chapter named for him. However here we are only given things like dimensions and appearance, and not anything to identify individual ships.

Just as a sort of anecdotal point – Some years back I went to a Zheng He exhibit in Nanjing, at the back of Chaotiangong. They had replicas and measurements and the history, but I also don't recall ever coming across anything like the importance given to ship names in America (the Mayflower, Columbus' ships etc). The names of the ships were clearly of less significance as an historical feature, and instead people seem to have been more interested in the giraffe. I can't imagine an exhibit to the discovery of the Americas not mentioning Columbus' ships by name.

As to you point about names beyond that, an illustrative example would be the name Sun Qian, of Three Kingdoms fame, gave to a ship he built. He named it Chǎngān 長安, Eternal Peace. This is still very much in line with the naming convention a dozen centuries later in the Ming. But they weren't all of this type. Other names of the period included Fēiyún 飛雲 (flying cloud), Gàihǎi 蓋海 (cover the sea), Chìlóng 赤龍 (red dragon), Chímǎ 馳馬 (galloping horse) and Dàbó 大舶 (literally "big ship"). These are all Three Kingdom's period ship names.

Would Zheng He or the Yongle Emperor even name their ships? Would a Tang dynasty riverboat captain name his ship? A fisherman in the Yuan period?

I can't say for a fisherman in the Yuan. If such records exist they're buried in Gazeteers I don't have access to. But I could imagine it would be similar to how it is today where larger ships get names and the pontoon boat that Uncle Cletus does his sunfish fishing off doesn't. Certainly that pontoon boat isn't making it into the historical record.

But a riverboat captain is more likely. Anything likely to be hitting ports are a safe bet. Remember that these Three Kingdom's period ships are all mostly river boats, and not deepwater ships. A name like Eternal Peace or Calm Seas could be a very auspicious thing. But again, such things would be buried in the Gazeteers of local government officials. Were I still at my previous institution I could go dig them up in the extensive collection in the library basement. Unfortunately that's not an option right now.

Did ship naming conventions vary by region due to the vast number of ethnicities and cultures in China? Would a Liao merchant name his ship but a Kashgari Uyghur on the Tuman not?

Generally if it was a Sinified group to the extent that they were using Chinese characters, then don't expect much variety. Remember that the various non-"Chinese" groups like the Yuan and Qing took on Chinese names and were very much within the linguistic sphere of influence. I couldn't tell you about Kashgari ship naming practices.

I'll see what I can dig up in the few records that have been digitised, but mostly they haven't and are only available in restricted physical collections.

Anyway, hope this helps at least get you in the right direction.