The Employee View
No, you do not.
Companies often want you to sign things at termination (noncompetes, IP agreements, NDAs, promises not to sue them...). These agreements are ubiquitous especially in tech jobs in the US. You are asked to sign them at employment, which you have to do or not get employed. Then you are asked to sign them at termination (or in your case, way after termination), but you don't have to, because in this case they are not offering you anything meaningful to do so.
Usually if they are serious about having you sign it, there will be a severance or other payment associated with the agreement that lets you justify the possible problems from signing by compensating you with real money in hand. This "consideration" (legal term) is also an important part of whether some aspects of contracts like that (especially noncompetes) are enforceable in some jurisdictions (here's a link to the rich set of rules around noncompetes across some of Europe). But if they are not offering anything, and it's after the fact like this, just ignore their attempt to contact you.
It does not matter if “maybe it’s legally unenforceable.” You still don’t sign it because there’s no reason to enter into a legal agreement for no reward. And even if there is a noncompete, it’s sure not your job to tell them where you’re working now. Let them figure it out themselves if they’re trolling to see if they should take action against you. You're not working for them any more, what are they going to do, fire you?
The last place I left, I gave my notice and they said on my last day "oh don't forget to sign your departure paperwork." I said "Sure, let me look it over." As is customary in the US for tech jobs, it had all this stuff in it (IP warnings, noncompete, nonsolicitation, "you're not going to sue us"). But there was no consideration involved (I didn't get anything from signing it), and I am pretty uninterested in working for that place ever again. So I didn't sign it or send it in. They didn't even bother to follow up about it. (It was only restating the existing IP and NDA and noncompete stuff I'd already signed.) If anyone had contacted me about it, I'd politely say "I'm sorry, I don't enter into legal agreements for no consideration. Thanks, have a good day!"
A previous place I left, they gave me $1k to sign a termination agreement, so I signed it, I mean, I didn't plan to steal their IP or sue them and that's a thousand bucks in hand so why not.
The Management View
Full disclosure - I’m a manager who hands this kind of thing out myself to terminated employees as part of my job.
This is usually containing terms they’ve agreed to already, as a reminder and “acknowledging it again” so they can’t claim in court “Oh I don’t know, I just signed some random things when I was hired, I didn’t know I couldn’t take my code and their customer data with me when I left”. Just giving it to them again whether they sign it or not is material legally because they can't claim they weren't told. Sometimes it's also a "you're not gonna sue us, arbitration, etc etc." agreement. Companies do have to protect themselves, and as even some questions on this Stack indicate people feel free to lie, take code, take sensitive information, and so on (and the vast majority obviously don't read the papers they sign or their employee handbooks or whatever). There's no 100% way to prevent it, but that's where legal remedies come in.
Ramifications Of Not Signing
I just had someone sign one last week, but I paid them two weeks severance in the bargain, so they willingly signed. Otherwise, I’d ask them to sign it but have no real recourse if they didn’t. Legally I can’t withhold PTO payout, or any other kind of coverage or thing they're due if they don't sign. I can’t smack talk someone on a reference check for fear of legal problems too.
If I feel strongly about it (especially if they were rude or unprofessional during the interaction) I can not actively help and recommend them and not consider them for re-hire, which is definitely a downside for them unless they are being fired for severe cause, as I try to help out people even if they didn’t work out on my team, and I’m active in my city’s tech community. If you quit or were laid off or were fired from my team on good terms, I will have your back in the future as long as you behave professionally.
If you tell me "I'm not going to sign that without consideration" - I personally sympathize, and I will try to get something for you, which will vary by how much my organization really cares. I wouldn't try "$1!" or whatever, that's a transparent legal dodge not suitable for real professionals. But it'll probably be modest, like my examples of $1k or a couple weeks pay, if you're not an executive or something. In reality this entire exchange is usually just a polite formality and a small sweetener so you're not just "signing because you loved me as a boss." If you try to negotiate it up from there ("two years!!!") I'll decline, and also consider you to have poor judgement as a professional (see above paragraph about not recommending you to others). I may also keep an eye on you more in the future to see if you do seem to be violating your employment agreement.
Ramifications Of Actually Doing The Bad Things The Document Is Talking About
Keep in mind most places aren't trying to arbitrarily screw you, they're trying to preserve their legitimate business interest. If you're working for a company working specifically on thing X, and you go to another one working on nearly identical thing X, you are very likely to be leaking specific information about product internals or customer relationships or whatever your deal is. That's why IP agreements and NDAs and noncompetes exist. And whether you signed one going out the door or not, you probably signed one coming in.
I would only take action if I felt there was a compelling business threat. But if you leave my, say, startup doing educational scheduling and go to our competitor who's the one other startup doing educational scheduling, and they get the benefit of our confidential internal tech and relationships and whatnot, I will drown you in lawyers. And to be blunt, even if if the contract is eventually held to not be enforceable, you're the one who's going to be spending a lot of your personal money on lawyers you'll never get back. So don't do it.