Is one allowed to make a similar game or even a sequel which would usually require obtaining a license? Who keeps the license in the end, i.e. who is to be asked for permission in such cases?
The basic answer is that in 90% of the cases, it is essentially impossible to figure out who owns the license, due to it possibly being sold to a string of liquidators. This is where the concept of Abandonware originally came from, and there are tons of early PC games with completely unclear ownership. To make any sort official sequel you will have to track down the current rights holder, which may very well be absolutely impossible.
If you put some effort into tracking down an owner and cannot, it is very likely that the current owner is not aware that it owns a certain property or does not care. In this case it may not be "legal" to reuse aspects of the game but there is essentially 0 chance of any legal issues. However, if your clone/sequel manages to become a financial success be prepared for the rights owner to show up one day and demand a huge cut of the profits.
Also, making a similar game, even a spiritual sequel, is unlikely to actually be actionable in court even if they do show up. Reuse of any specific asset (art, music, text) is very easy to prove as illegal infringement, but doing a gameplay clone of an abandoned property is generally safe, based on the thousands of such PC games in existence.
In addition to Iain's and Ben's very good answers, an important concept is the differentiation between "what is legal" and "what is enforceable."
Just because no one has a particular interest in enforcing their Intellectual Property rights doesn't mean they don't own them and couldn't suddenly change their mind about where to put their legal focus. It's safest (and most ethical) to make an honest attempt to seek permission from the rightful owner. This may involve cost to you, as to hire someone to find the rightful owner.
Of course, you can choose to "live life on the edge" and just go for it, but the legal/right thing to do is to put in the effort.
Btw, I have some experience with asking for permission to clone what you might think of as abandonware, and the rightful owner telling me "no problem, we're totally done with that IP, now", so don't think it can't happen.