You might be able to deduce the server and web host via the WHOIS information. You might also be able to guess the hosting type (shared, reseller, VPS, dedicated, cloud, etc.).
Clues from the Name Servers
If they use the web host's name servers, that will (usually) tell you what web host is being used, especially if the name of the host is in the name server's domain name.
But, people are generally not required to use their web host's name servers. They could easily use the name servers at their domain name registrar or even another web host, and point the domain to a different web host. They could also use private name servers using one of their own domain names.
So the name servers they use may give you a clue, but maybe not.
Clues from the IP Address
The IP addresses that their domain is assigned to (pointed at) could also give you an indicator of the web host, or at least the data center or upstream provider. But, again, knowing this information won't tell you what hosting plan they have nor the price they are paying.
Now, if you put some of this information together, you might be able to guess.
For example, if the name servers tell you they are at a particular hosting company and/or the IP address of their website belongs to a certain web host, and you can determine how many other websites are using that same IP address, you can guess the hosting type.
If there are hundreds or thousands of sites from different owners on the same IP, they are probably using shared or reseller hosting (or they have a dedicated server and sell hosting to others). If there is only a few sites on the IP address and/or all the sites at that IP are the same owner, they probably have a VPS or dedicated server (or are on shared or reseller hosting and have a dedicated IP address).
Add in what you know about the website to narrow down the possibilities. If the website is large, complex and/or has lots of traffic it is more likely to be on a VPS or dedicated server. Whereas a website that is small, simple and doesn't have much traffic would most likely be on shared or reseller hosting. If they have an SSL certificate on their site, then they have a dedicated IP address, which could be on any hosting type.
Not super specific, I know, but if you combine the information about the possible hosting type and information about the possible web host, and you could guess the hosting plan and pricing. But it's a guess.
Information that Obscures Information
And your guess could be defeated by a combination of private name servers, dedicated IP addresses, using the domain registrar's name servers, using the name servers at a different host, etc. Another snag would be if they are using a web host that is reselling space on another web host.
So you won't really know based on just the WHOIS information.