Docker extension adds "Build Image in Azure" in right-click menu when I have no Azure settings or extensions

This is by design, and given the extension itself is maintained by Microsoft, it shouldn't surprise you that they've baked in integrations with Azure out of the box. The extension's package.json file defines it to be shown without conditions when installed:

// <snip>
{
    "command": "vscode-docker.registries.azure.buildImage",
    "title": "%vscode-docker.commands.registries.azure.buildImage%",
    "category": "%vscode-docker.commands.category.azureContainerRegistry%"
},
// </snip>

There is no way in VSCode currently to disable individual commands from extensions that you are not the owner or maintainer of. If this is really that bothersome to you, you could fork the extension on GitHub, remove the command from package.json, repackage it into a *.vsix, and add it to your VSCode installation manually. This is generally more trouble than it's worth, since you would then necessarily have to pull upstream changes from the main repository periodically to make sure you have the latest bugfixes and security patches.