In the US, in general, universities do offer the student a fancy paper diploma when they complete a degree. This is such a strong tradition that I would be pretty surprised if there are any universities that don't do it at all. I don't think that JeffE meant to imply that there were.
However, in this culture, this diploma is usually considered to be purely ceremonial in function, and isn't normally used as proof of the degree. For one thing, diplomas usually don't have meaningful anti-forgery security features.
And there are various circumstances in which a student who had completed a degree might not have a physical diploma. Maybe they didn't attend their graduation ceremony, and the university didn't have their current address to mail it; or maybe they used to have it but lost it at some point. Since it's a ceremonial document, some people might not go to the trouble to obtain or replace it.
As such, in the US, if Alice needs to prove to Bob that she holds a certain degree, the standard method is that Alice sends a request to the university that they send Bob a copy of her transcript, which not only records her degree but also her grades in all the courses she took. This will normally be printed on security paper, sealed with an official seal, and mailed directly from the University to Bob, so that Alice has no opportunity to tamper with it. (There could still be ways for Alice to have a fake transcript sent to Bob, but it's harder, and so this process is usually considered adequate.)
For privacy reasons, Alice's transcript can only be sent upon Alice's request (and she'll have to authenticate the request somehow, e.g. by giving her university student ID number which isn't normally known to anyone else). Bob can't get it without Alice's cooperation.
JeffE may have overstated the case when he said this was the only way to prove it, but it's by far the most common in the US higher education system, and other forms of proof are much less likely to be accepted by Bob.
(I think JeffE was also objecting to the term "PhD title" since this term isn't used in the US to refer to a physical document confirming a degree. So if you asked someone for their "PhD title", they would not know what you were talking about.)