Self-identify just means you haven't had to provide evidence upfront. There can be objective criteria, but you haven't been asked to provide it and are trusted to have told the truth until there is evidence to the contrary.
What objective paperwork would people like for ethnicity? What you look like doesn't count as official documentation.
In the US and Australia people can classify as indigenous even if it was one person in their family tree five generations or more ago which I always find bizarre as that person would presumably not have experienced any discrimination due to their race that far removed from that ancestor, so why should they get preferential treatment?
In the US, individual indigenous nations decide now, just like any other nation decides the criteria for citizenship. That does not automatically mean 'preferential treatment' (not sure what treatment you're talking about, but not all American Indigenous nations are federally recognized anyways). Even though not everywhere requires proof of enrollment, random people making a claim doesn't really do much without that.
Blood quantum - which is what you're talking about and a very controversial topic as it's what governments applied to many nations (and was largely done by self-identification at the time) - is still in use by many, others use old tribal rolls instead, some use both, and they can use whichever they want. Don't know any that use hardship as a defining criteria, but then that's not the defining characteristic of any Indigenous culture. There is a lot more to us than that.
I don't need a certificate to ask someone if could sit down, I sure as hell do if I want access to restricted parking or limited access items.
Blue Badges, yes.
RADAR keys, which are often thought of as a limited access item - no, anyone can get one, no one asks for any evidence. It's generally considered more a detriment to put in more hoops there, or with mobility device use...