Is it ethical to privately thank authors when you discover they’ve cited your work? Can it be looked upon like you’re trying to gain future favour? Or like the citation itself was for any reason other than purely academic reasons?
It's not really appropriate to thank them for citing you, which is something they are obliged to do if your work is relevant and shouldn't do if it is not.
What may be appropriate is to thank them for taking the time to understand a very important point and for how well they've summarized your work and note that they've given you a new insight into it (or how it relates to other work). That is thank them for their work and insights.
Normally the point of an email in such a case would be to point out a more recent work that they've missed (of yours or someone else), or to point out that there's a subtlety that they overlooked or (more humbly) to question why they didn't use that approach or used a particular alternative, or asking them to comment on some followon work that you are doing or proposing.
Actually, this last is the most common time people contact me about my papers, to ask for comment or help on something they are trying to do or proposing to do. And sometimes it can be an invitation to collaboration of some kind.
It is a good idea to start a dialogue with people that are working on similar things and have a compatible approach - getting to know the people in the field is important for matters like conferences, grants, examiners, sabbaticals, etc. It is particularly important in times where travel is more constrained.
Is it ethical to privately thank authors when you discover they’ve cited your work?
I don't think it is a question of ethics, but would be weird.
Can it be looked upon like you’re trying to gain future favour?
No, but I would view the email as unusual, but not trying to gain favor.
Or like the citation itself was for any reason other than purely academic reasons?
By who? Both you and the author know the citation was only based upon merit (or the original author's motivation).
Instead of thanking, just drop a the authors a note saying you found their recent paper interesting. Perhaps something like:
Dear Dr. Smith,
I saw your recent paper, New Widgetology updates. I was impressed with your advancement/progress/work and how you build upon existing work/theory/etc.
< Add a comment, question, invite to collaborate, have a Zoom call here>. user354948
That will let them know you read their paper.
There is an elephant in the room, and I feel you are not addressing it.
The citation in itself has no intrinsic intellectual value. You may be flattered that your person has been cited—but in reality it is your work (most likely with your coauthors), not your person.
Therefore, why would you be thankful? If it is because you found the citing work interesting, then you should say thanks to all the interesting papers you have read, not only the ones citing you.
If it is because you see your citation counts growing ... well, keep in mind that the "real" worth of citations is logarithmic: it makes no difference if you have been cited 2 or 5 times; the real difference is between having 10s or 1000s of citations. And even that difference is simply stating "how many research slaves (phds, postdocs and the likes) have the author had", unless the author is a world-famous expert on the topic (a status you do not reach with citations).
If you cheer each single citation you receive, you are (involuntarily and as a side-effect) feeding the citation gaming.
Additionally, you are enforcing (again, involuntarily and as a side-effect) the belief that citation ranking have to mean something and that the citation-based metrics have some intrinsic value (therefore feeding the citation games).
Finally, if I would receive such a thank mail, I would think that you are a naive young researcher that believes in meritocracy, that understand nothing of the power balance in research, that do not understand that the road to hell is paved with good intentions and I would reply to you asking "why?".
Disclaimer: I am out of academia since years, and my h-index is 19 (yes, pun intended, citations and related metrics have no intrinsic values. I am just mentioning it for your pleasure).
I strongly advise against this.
There seems to be this fallacy that everything in academia is ''ethical'' or ''not ethical'' and that's all that matters (similar to the fallacy of people learning a language who divide things into ''grammatical'' and ''not grammatical'' in that language and don't appreciate that something can be grammatical and still sound strange).
Yes it's ''ethical'' in the sense that it's not wrong, but it's most definitely not a normal thing to do in academia and would look strange. It is not normal practice and it is just ''white noise'' to clog people's email inboxes with random pointless emails like this. If someone sent me an email like this, I would just delete it and be vaguely annoyed that I had to spend a bit of time reading and deleting it.
A simple "thank you" unethical? OMG how did we get to this place?!
It would be perfectly fine and not at all weird.
As long as you are simply thanking them for acknowledging your work and not making a big deal about it, it's simply a normal, decent thing a person can say to another person. We are after all still human beings.
If you add some additional chit-chat or mention something related to research they might find of interest, or "while I've got you here, I'd like to as a question" it may help to make your message feel more normal to you and to them, if you feel it's a concern.
Academicians should not live in balkanized ivory towers, they should be able to feel a sense of community and camaraderie1, and how better to do so than to treat each other in a warm and social way.
How can a simple "thank you" be anything but good?
Beyond that, you may make a friend, or find further direct discussion useful, or even collaborate some day.
Consider it an ice-breaker; go for it!
1though it may vary by field, there really used to be much more of this decades ago than there is now, and while some folks feel cut-throat competition brings out the best in everyone, I'm not so sure.