Ban after 3 months

This happened to me once also. The answer is number 2 with support's assistance.

First contact, if not responding or willing to cooperate, then send ticket, suggested to use the category: other.

Do I need to add a Tank Game to my Steam account (the one that this site didn't find on my account)? Or will its presence / absence now affect nothing?

It will affect on sgtools checks in some GAs (you can find them in discussions). So it'll be better to add this game to your account.
https://www.sgtools.info/nonactivated/Abstinenzler

I would leave a comment on the giveaway letting the giveaway creator know what happened. If they are able to provide a key for the correct game, that can resolve the situation, but most likely they do not have a key and won't want to try to get one because they just selected the wrong game when creating the giveaway.

I would then ask the giveaway creator to create a support request to delete the giveaway.

If you cannot get in contact with the original giveaway creator for some reason, create a support ticket yourself. You can always create a support ticket at any time to ask support to remove the "received" feedback that you left on the giveaway so that you no longer have an unactivated win on your account. That will make your account look better if someone checks your account with sgtools because the unactivated win will not show anymore and also, I'm not sure how it works, but support can probably leave a note on your account explaining that the suspension you received was because of an accident and you didn't intentionally break the rule.

Make a ticket to request not receive feedback. Then contact the giveaway maker.

the most useful advice I got on this forum is this:
every time you win and activate the key you receive, check your own profile through sgtools like so -> https://www.sgtools.info/nonactivated/Abstinenzler
if after you activate the key you receive sgtools shows your win as red, you should probably be alerted that something is wrong (e.g. two games with the same name, like your case)

Does it make sense to write about this situation in Support SteamGifts.com or does the Support not respond to user requests (3 months ago I sent 2 different requests there, I still haven't received any response, tickets are open)?

Staff prioritizes different ticket categories differently. Don't expect prompt replies (or even ones within the same year) to Add Game to List or User Report tickets. If it's any other category (and not in relation to a giveaway with multiple copies, which would specifically need Super Moderators to handle), that'd be a bit unusual.

Do I need to ask the author of this giveaway to delete his giveaway after 3 months from the date of completion, since the key was not from that game? Or will it give me nothing in the future (for a second similar mistake in the future, I should already be banned forever)?

No, you shouldn't do any of that. It's not your responsibility to bear that- nor, even if it was, would such limited Not Activations endanger your account in any way, on their own; Rther, staff makes determinations based on excessive rule infractions, either by way of extensive infractions over a long period of time or by many infractions within a short period of time. Rule of thumb seems to be something along the lines of (nothing but infractions on a new account), (roughly 5 infractions in quick succession), or (roughly 10 infractions in total) before staff action is far more likely than not. [Ultimately, it boils down to the staff member considering the matter, so the exact numbers will change based on that.]

( Note also that even if staff does take action, it'd be an indefinite suspension and not a ban- bans only occur if you harass staff during a suspension. You can still get your suspension removed by communicating with staff (which you can still do during a suspension) and arranging to activate some or all of the games you have infractions on. )

Rather, the best course of action for this matter is to note it to staff via an Other ticket (though that may be unecessary, now that that this thread exists). The most likely outcome would be for the feedback of the game to be changed to Not Received (as it's the GA creator's obligation to provide the game listed, and that fact will generally trump any obligations you would have in this particular type of situation).

However- and especially if you indicate satisifaction with the game provided, given the fact that it's an exact title match (and therefore may be an honest mistake)- staff may instead lean towards changing the giveaway listing to match the provided game, if they feel that would be more suitable for the circumstances.

Further, note that deletions for giveaways only occur if both parties involved agree to the deletion (or if only one person has any say on the matter, as per a GA creator who has a winner which is eligible for reroll/declined their win/etc). As such, you have some degree of power as a winner when it comes to deletions- that is to say, you're not typically obligated to provide approval for them. Most of the time it's polite and respectful to agree to such, so I'm by no means encouraging otherwise; However, do note that if you ever feel it more suitable for the GA creator to instead provide the game they promised, you do have the option to refuse a deletion request.

Again, not something I can see a real merit for (especially if you've already gotten a game out of the matter) but, at the same time, I don't want you to feel like you're in a position of primary responsibility, as your phrasing appears to indicate you believe. As the matter has been presented by you (and without accounting for any special considerations staff may or may not have in regards to the delay in your bringing up the matter to them) you should be fully in the clear*, as the onus should be fully on the GA creator for mislisting the giveaway. Rather, SG is usually winner-centric in its rule bias (sometimes to the detriment of GA creators), so it's not all that often you'd be in trouble for a matter the GA creator also has a part in.

* [Rather, the temporary suspension you faced is actually the punishment the community typically associates with GA creators who don't check their wins, so from that perspective you've already "served your sentence".]

Do I need to add a Tank Game to my Steam account (the one that this site didn't find on my account)? Or will its presence / absence now affect nothing?

Non-Activations are automatically determined by the site checking against your account during syncs. Thus, adding the game to your account would automatically resolve any issue (as would the deletion of the giveaway in question, as the check would no longer be performed from SG's end). Thus, if the onus was ever on you to resolve a situation, then activating the correct game or arranging a deletion with the GA creator (often rather easily done just by compensating with an equal or higher value game) would indeed handle the matter suitably. However, as I noted, you likely don't have any actual responsibility in this matter (though, ultimately, such a determination would be for staff to make).

Ideally, you would ask the GA creator to submit a ticket to "change giveaway game". Ideally.
I was able to get one of my GA games changed before with that method, but not sure if response time would be an issue for you.

Happened to me https://www.steamgifts.com/giveaway/WSMwc/pumped-bmx , I won but it was for the game Pumped BMX Pro instead of BMX+ ... I never actually got suspended though(maybe I caught it in time?,) I just noticed it months later when I checked my activated wins on SGtools. Since the creator was rarely logging in and has a good history I opted to just buy the game, didn't feel fair to bother them after so long...support wanted me to "double confirm" in writing that I was indeed going to buy that too,I think they follow-up and make sure. Luckily it went on sale -90% the next week.

If people check you on SGTools and see that you have that non-activated win they will get angry.