My question is whether it is possible to group the updates by computer/server? Since I can't find the possibility. This is because I want to, for example, install all updates on a specific server and not on
others (which are also located in the same group).
This group (target-group) setting cannot control selective installation of patches in computer/server having the same client-side targeting. If you've approved X updates for this particular target-group, all the updates relevant to the OS approved in the set of X updates will be downloaded by client system when it communicates to WSUS the next time.
The better option for your case would be to create different target groups for different types of servers/desktops. Then you selectively install few updates on a few servers (in the first group), and deploy all the updates pertaining to other servers/desktops (in the second group) completely. You'll have to divide the set of systems into 2 OUs each with distinct GPO setting as described below.
This setting of client-side targeting can be done in AD by creating 2 separate OUs and using 2 GPOs (one for each OU) to achieve this:
(i) first GPO setting with the target-group set as selective, categorise the selective updates required systems in this separate OU and deploy the selective updates using WSUS,
(ii) the other GPO which you apply to the 2nd OU, categorise the systems marked to receive all the updates in this 2nd OU (having remaining servers) and GPO setting having second target-group.