If the purpose of a registered study changes, is it still a registered study?
First off, as you state it your worry is as follows:

Given the divergence of purpose, should the paper make the claim that the trial was registered?

Unpacked and clarified, I think this worry contains the following assumption:

A paper should only claim that it is a registered trial if the purpose of the trial that is registered and the claim that the paper looks at are closely connected.

I think this assumption is basically false and misunderstands the purpose of registration.
Looking over clinicaltrial.gov 's explanations of why and when to register trials, they suggest five benefits of registration and five benefits of the data, one of which is:

Facilitate systematic reviews and other analyses of the research literature

Several of the other things they state are similar in function. (There's also public benefit, ethical review, avoiding bias in the use of the data, and other features).
The data from Clinicaltrials.gov is free for anyone to use subject to these terms and conditions. The relevant item is :

In any publication or distribution of these data, you should:
Attribute the source of the data as ClinicalTrials.gov
Update the data such that they are current at all times
Clearly display the date the data were processed by ClinicalTrials.gov
State any modifications made to the content of the data, along with a complete description of the modifications

Thus, if you use data from Clinicaltrials.gov to produce research, you must cite that you took it from there and presumably should say which trial you are working from.
The specific method to correctly cite would be a conjunction of these terms and the journal's policies for citation.
On a general level, there's nothing ethical wrong with saying where the data came from. In fact, it's an obligation.

Returning to your final question, I think we can nitpick them for their English a bit. The authors might be better suited using a different phrase than

This trial was registered at clinicaltrials.gov as NCT00080171.

which they use in this paper and another publication where in the full-text they state:

Data for this trial (NCT00080171) were obtained from the Osteoarthritis Initiative (OAI) database.

This leads me to believe they are using &quot;trial&quot; in a somewhat odd way (at least from the perspective of a native AmE speaker who works in the humanities).
Something like:

This research is based on the registered trial NCT00080171 from clinicaltrials.gov.

seems better in both instances.
But while nitpicking for English, I think that they must state that trial they used is registered but shouldn't make it seem like this was that trial's purpose.