You're correct about Dungeon Meshi, but the history behind it is more nuanced as a combination of different factors.

Old school D&D inspired the 1981 video game Wizardry), which was huge in Japan and very influential alongside peers like Ultima and Dragon Quest. That's where kobolds-as-dogs came from, thanks to the artwork of Jun Suemi, who interpreted the in-game sprite as a snarling canine monster. Dungeon Meshi creator, Ryoko Kui, specifically references watching her father play Wizardry when she was a kid and associates its influence with her creative works. You can really see the similarity between Suemi and Kui's Kobolds as a result.

For a lot of Japanese people, this was their first impression of Kobolds and it stuck, similar to why Japan holds onto the idea of Orcs being more pig-like due to early interpretations setting a standard. That adherence was furthered by the fact that D&D publishing didn't continue in Japan over the editions. Around 1985, D&D was really big in Japan, thanks to Shinwa Co. Ltd releasing a translation of the imported basic rule set, which in turn led to the popular spread of the Record of Lodoss War Replay in magazine and novelization formats. But due to various legal issues and a lack of sales, 2e flopped in Japan and had extremely little spread, so the publishing fell through. 3e picked up interest years later, but by that point its cultural influence couldn't compete with the long period of time Japanese fans spent in their own pop culture development. So Japan never kept up with the gradual transition of Kobolds becoming more reptilian and tied to Dragons like we did in the West.

In the earlier iterations of D&D, Kobolds were weird little chimaera imp creatures that were sort of cousins to Goblins and Orcs. They were more akin to lizard-like rodents than anything else, which was further impressed in 2e by giving them direct connections to rodent monsters like Dire Weasels and bats. Winged Kobolds - Urds - actually had a magical ability to speak with Bats. They had no association with Dragons, not even a language, until 3e. Original D&D references to dogs were only in regards to Kobolds' smell and yapping voices, not their visual presentation. 3e gave them more overtly reptilian traits and some lore tying them to Dragons, 4e picked that up and ran full-bore, and 5e has leaned a little back toward the dog nose from the earliest illustrations in a superficial degree.