As far as I am concerned, I don't know personally someone who is or was in the same situation as you. However, being 4 years in in my own Ph. D. and knowing a lot of people who are doing one or finished one, I can maybe bring some perspective to your situation.
Among people I know and including myself, I've seen two typical situations upon starting a Ph. D.:
either the student doesn't know exactly the topic (s)he will work on and only got a broad field to investigate,
either the student is actually deepening a topic (s)he worked on during a master thesis (this is my case).
The first situation seems fairly common in my experience and is arguably worse than yours. Not knowing what you will be working on for the 4-5 next years is probably more disheartening than having a topic that is close to your interest but not exactly matching what you had in mind at first. Sometimes, people change of topic every 6 months (because they didn't find anything with the previous topic they were working on) in the beginning and don't seem much more advanced 2 years after starting to work on their Ph. D. thesis. In fact, most people I know who stopped their Ph. D. started off like this (note that this isn't a rule of thumb; I've seen people in the same situation finishing their Ph. D. as well).
The second situation sounds, of course, ideal. You start your Ph. D. already knowing what to do; sometimes you even get to publish what you worked on during your master thesis right from the start or by slightly re-working it/expanding it (depending on how your master thesis was received). However, this doesn't necessarily shield you from other issues you can encounter while doing a Ph. D., such as:
- having trouble to explain to friends and family what the heck is your thesis about (don't underestimate this problem, and if possible, train yourself to explain your topic in the simplest way),
- getting sometimes the feeling that your research doesn't really interest people in your own field (because people keep asking you during conferences what's the endgoal of it, because your paper submissions get rejected because reviewers have trouble understanding what you're doing, etc.),
- doubting about whether you should finish or quit your Ph. D. when you're halfway through it, even if you managed to publish some papers (a phenomenon which has been dubbed as "the Valley of Shit"),
- having irregular schedules (except if your university enforce them), sometimes working all the time (even the week-end) for some weeks and sometimes not working much because you're unsure about what you will do next (or because you want to slow down a bit), which can also be a problem on the social side,
What I'm trying to say with my points above is that doing a Ph. D. is a difficult task anyway, even if you manage to get the best possible topic, and it's perfectly normal to feel unmotivated at some point. Doing a Ph. D. isn't just an intellectual challenge.
Also, be aware that the way you progress in your research can lead to developments you wouldn't have expected at first. In my case, for academic reasons, I had to write a kind of big planning for the first years at the start of my Ph. D., but I can guarantee you it looks nothing like what I actually did up to now. I don't know much about your research topic, but you can already keep in mind that you might not always work exactly on what you expected to, and you might even reconnect with the topic you first had in mind or go into completely different directions. I hope this helps.