The ePrivacy Directive
Cookies are governed in the EU by the ePrivacy Directive which was first passed in 2002, and revised in 2009. However, if a site places a cookie, and later reads the cookie and stores an indication of the cookie content, this may be personal information and thus also subject to GDPR rules.
(Other privacy laws, such as the CCPA, may also impose requirements on cookie use, when they apply.)
Note that the so-called "EU cookie law" is a directive and not a regulation. This means that it is up to the various EU countries to implement it in their national legislation, and different countries may implement it in different ways. It would be the actual laws in individual EU countries that would be binding on website operators, and I am not going to try to find and analyze all the various laws on the topic.
Paragraph 25 of the directive (linked above) reads:
However, such devices, for instance so-called "cookies", can be a legitimate and useful tool, for example, in analysing the effectiveness of website design and advertising, and in verifying the identity of users engaged in on-line transactions. Where such devices, for instance cookies, are intended for a legitimate purpose, such as to facilitate the provision of information society services, their use should be allowed on condition that users are provided with clear and precise information in accordance with Directive 95/46/EC about the purposes of cookies or similar devices so as to ensure that users are made aware of information being placed on the terminal equipment they are using. Users should have the opportunity to refuse to have a cookie or similar device stored on their terminal equipment. This is particularly important where users other than the original user have access to the terminal equipment and thereby to any data containing privacy-sensitive information stored on such equipment. Information and the right to refuse may be offered once for the use of various devices to be installed on the user's terminal equipment during the same connection and also covering any further use that may be made of those devices during subsequent connections. The methods for giving information, offering a right to refuse or requesting consent should be made as user-friendly as possible. Access to specific website content may still be made conditional on the well-informed acceptance of a cookie or similar device, if it is used for a legitimate purpose.
The notice described in the question neither describes the purpose of cookie placement, nor does it offer any meaningful choice to accept or reject them. As such, I do not think it would be compliant with the directive. If cookies are read back (which is after all pretty much the point of having them) and can be potentially identified with an individual, then under the GDPR (article 6) there must be a lawful basis for processing them, and the interaction above would not be enough to establish consent as a lawful basis. There might be some other basis that does not require consent, however.
Interpretations from others
This page from cookiebot explains some of the history of the directive, and how it contrasts with the GDPR (which is a regulation, and so is directly applicable without the action of national legislatures).
This page from The Verge discusses recent changes in the directive and the guidelines for applying it, and criticizes how it has been complied with, and ignored, by many sites.
Cookies and the GDPR
This page, "GDPR, and the ePrivacy Directive", from GDPR.EU says:
... cookies, insofar as they are used to identify users, qualify as personal data and are therefore subject to the GDPR. Companies do have a right to process their users’ data as long as they receive consent or if they have a legitimate interest.
Passed in the 2002 and amended in 2009, the ePrivacy Directive (EPD) has become known as the “cookie law” since its most notable effect was the proliferation of cookie consent pop-ups after it was passed. It supplements (and in some cases, overrides) the GDPR, addressing crucial aspects about the confidentiality of electronic communications and the tracking of Internet users more broadly.
To comply with the regulations governing cookies under the GDPR and the ePrivacy Directive you must:
- Receive users’ consent before you use any cookies except strictly necessary cookies.
- Provide accurate and specific information about the data each cookie tracks and its purpose in plain language before consent is received.
- Document and store consent received from users.
- Allow users to access your service even if they refuse to allow the use of certain cookies
- Make it as easy for users to withdraw their consent as it was for them to give their consent in the first place.
That page also describes the proposed ePrivacy Regulation which, if passed, will replace the current directive, and links to drafts of it.
The site described in the question probably does not comply with legislation implementing the ePrivacy directive. Depending on where it is hosted and targeted, and what it does with cookies that it sets, it may fail to comply with the GDPR as well.
There has been very little enforcement of the directive so far. This may change in the future, particularly if it is replaced by a regulation, as has been proposed.