If the tag is a pure number, does it have a "~" before the number in the url of its wiki page?

For example "~3886"

yes, see wiki_page.rb:
https://github.com/danbooru/danbooru/blob/af7c40d98e537ade6a9fc3e8bdb7793977eb72dc/app/models/wiki_page.rb#L29
it performs an ID lookup if the value consists of only digits, otherwise performs a title lookup -- normalize_title strips the ~ prefix if present, so that prefix is used to force a title lookup.

You can actually confirm this really quick using DText.

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