I did find a German news article from 2016 saying that Angela Merkel is an Estonian e-resident - presented to her by the Prime Minister of Estonia. So it seems plausible, but that it's something presented to them as a symbolic PR stunt.

"On Thursday, Prime Minister Roivas will present Merkel with an e-residency card - allowing the chancellor to become a digital resident and try out Estonia's digital solutions firsthand."

http://www.dw.com/en/in-estonia-merkel-and-roivas-talk-russi...

There's an Estonian story with photos about that here:

http://estonianworld.com/technology/angela-merkel-becomes-es...

But she didn’t “sign up” for residency, they “presented” it to her which is a whole different scenario.

I don't think it's a different scenario. Even the Estonian press says Shinzo Abe was presented with his e-residency, rather than him applying:

https://news.err.ee/115597/japanese-prime-minister-becomes-e...

"Abe's e-residency card was presented to him by Taavi Kotka, the Deputy Secretary General at Estonia's Ministry of Economic Affairs, currently on a visit to Japan.

Kotka told ERR that e-residency was Estonia's gift to Abe, which he accepted."