Is there a way to get auto-completion for tags:: page| in the way one gets completion for [[page|]]?
Don’t thing so. You can alternatively use # instead of [[]]—might be quicker.
I am using tags:: at the beginning of a page to specify all the tags (or other pages) that are related.
Do I achieve the same with a list of # or [[]]?
(I switched from # to tags:: a while ago)
Yes (though I suspect you might have misunderstood me). These are the same:
tags:: Tag1, Tag2, Tag2
tags:: #Tag1 #Tag2 #Tag3
tags:: [[Tag1]] [[Tag2]] [[Tag3]]
Only the latter two will get you autocompletion.
I’m not recommending not to use tags::! 
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