I wasn’t aware of the click to chat function, so I checked it out. On the Whatsapp website it gives clear instructions on how to place the link on your website. It’s a hyperlink that couldn’t be more obvious that your Whatsapp number is in clear text (and will show if you hover over the link, even if it isn’t visible as such). So on the face of it you are publishing your own personal information to the web, it’s obvious it will be visible to search bots, and there is no GDPR issue assuming you are using your own number.
The article and Many comments refer to a QR code/widget. This does not appear to be provided by Whatsapp – there was nothing came up on the Whatsapp website and search results were all for third party services. The article also notes that you have to go to a third party for the QR code. I just quickly made a QR code for my Whatsapp and there was no alt text for the image or other metadata. So if there is a data processing issue then it is with the individual’s use of whichever service they pick.
But underlying both these things are two questions: first, unless the link is being embedded in a user comment on someone else’s website, then the individual using the link or widget is putting it on a website they control. As webmaster (or at least content contributor), they have to take responsibility for the code. As you mention GDPR, then that person is also legally responsible for what PI they publish there, so if they put someone else’s PI (knowingly or not) then they are the one that is potentially in breach.
This leads to another question, which is what proportion of people are using this in an individual capacity (i.e. it is their personal number) and how much is for business use? I suspect the majority is for business use, and in that case then why would you be hiding your contact number? The article talks about identity fraud, but if you’re a business then all the information cited should be public domain anyway, and discoverable. And similarly from a legal perspective, posting a contact link that is obscured could be contrary to the Disability Discrimination Act.
I wouldn’t generally defend WA or any part of the social media commercialisation of privacy, but as many others have said, I don’t believe this is a Whatsapp issue. If there are genuine cases where private information is being leaked, then it points to people either lacking competence or failing to take responsibility for their own actions.