You seem to be talking about the right to be forgotten which exists only in some countries in Europe and only applies to sites within that jurisdiction. For that I do not know. As for the rest of the world, this is not a right and any request would be up to the discretion of the site owner. If you have a compelling case, and there are some, then I could see someone removing PID (personally identifying data) if the data source has different data.
Most of the time there is really no caching of whois PID data with very little exception. The reason for this is because of the massive amount of space whois (in particular) data uses even when compressed. I only know of one site that actually maintains historical whois: domaintools.com, though I am not sure that is available- it may be with a paid account. Most of the time, if the whois data is not changed and a site re-fetches the data, the PID will show up again and there is no mechanism to prevent that from happening. As a side note, most sites fetch data in real-time these days though not all. All it would take to refresh the whois data is to visit the page. Some re-fetch periodically. Some, very few, never refresh. With millions of pages, capturing whois data becomes more and more difficult and costly. It can take quite a bit of time to refresh the data even with the best of efforts. As well, some whois servers have restrictions as to the number of requests and so extra effort not to pound any whois server to death is often taken. It is actually, kind of a trick business.
Again, most webmasters and site owners will be glad to remove PID from this form of data if they can. It can be a simple process even if done manually. And most webmasters and site owners can understand wanting to divorce themselves from prior associations. They get it. If you want PID removed from a whois website, often just a visit to the page or an e-mail will do.