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VPN doesn't alter your machine or browser footprint just alters your exit node on the web, so it is possible they could still track you with those things. There are a few websites that shoe you how unique or generic your machine, os, screen res, browser etc.
Like if you are an iphone user with safari you are quite generic on the web, if you are using a rare phone, with a modded is, and an uncommon browser you will be narrowed down.
As far as I know, DDG does not track search results tied to users, however, your search terms are part of the URL, so your VPN provider could theoretically see them as they pass through their exit node, unless you are also encrypting your DNS requests.
DNS requests shouldn't contain search terms.
On some systems, DNS requests might expose the hostname of servers that serve sites for results that you're clicking on. Firefox and Chrome at least, probably others, defaults to use of DNS-over-HTTPS, so the hostname shouldn't be visible in plaintext even then.
Depending upon how the HTTP server is set up, due to SNI, it may be possible for the VPN provider to see the hostname to which you are opening a connection during the TLS handshake by looking at the HTTPS traffic. AFAICT, this is the common case today.
Sounded like they meant can DDG track their searches, which would be a yes, if they wanted to. VPN altering your IP doesn't stop search parsing, especially if your browser, os, screen red etc are somewhat unique then they'll be able to fingerprint the user
