I don't know, maybe they are colluding, but it is funny to default to that assumption over both platforms just delivering you the same content because you have the same behavior across both apps.
Notably, this started happening the day that I made my TikTok account public. My Instagram feed began to be a copy of my TikTok feed. The exact same videos. Even after changing my Tiktok back to private and deleting all of my followers, the feeds are still identical, every single day. My behavior is not. On Instagram, I follow and interact with very different accounts than on TikTok. It seems to me that Instagram is buying or accessing TikTok's data, and it is not through advertising providers, because the identical content is coming directly from Instagram/TikTok and not promoted ads.
There are three possibilities though. One is Instagram copying TikTok, without their knowledge, the other is Instagram copying your TikTok feed with their knowledge but not their blessing, and finally Instagram copying your TikTok feed with TikTok's knowledge and their blessing. If we take a look at http://TikTok.com/robots.txt, it seems if you make your TikTok public, TikTok is happy to let Instagram take a look at it (but not a number of AI crawlers). What Instagram does with it is up to them, but it's in robots.txt.
The simplest explanation would be that Instagram crawled my TikTok's account's followers and is curating my Instagram reels based on that point in time.
However that's not what happened, because my "following list" is restricted to be viewable by "only me", even though my account was public. "Public" just means that you can view my videos without me accepting a follower request. And I don't have any videos anyway.
So I can only deduce that setting it to "public" flipped some bit in either Instagram or TikTok's backend to where now they both are sharing the same or very similar data to curate my feeds.
In the book "careless people" it is highlighted that Facebook embedded spyware in their app which tracked other apps that were installed/used on the phone. This allowed them to figure out that so many of their users were installing WhatsApp, and enabled the legendary WhatsApp purchase. It is very much possible that Instagram is reading some sort of temporary files created by TikTok and extracts data using this method.
Facebook/Meta has a proven track record of fetching all data from your phone, even when abusing security vulnerabilities to do so. And the clowns at Apple can't even fix RCEs in their network-exposed applications, I'm not convinced the separation between apps is flawless.
Do you work on ad platforms? If you did, that imagination would be reined in quite a bit.
The gist of it is that the platforms carefully curate what data they provide to their customers. They don't want to give anything low level away, both for compliance reasons and because it would undermine their own business. Facebook makes money by charging for access to their users, not by giving away enough information about them for one of their customers to become a competitor.
Almost all of the data the platforms provide is 'How well is your marketing campaign doing and among whom' - on a high level. There are issues with poor designs leaking information when the campaign only hit 20 people, but as a general rule of thumb, but that's an exception, not the rule.
Not sure if they're explicitly sharing data, but there does seem to be something that's sharing data across the platforms. When I buy something from Tiktok, the ads for the same thing shows up on my instagram almost instantly. Doesn't necessarily mean they're directly sharing data of course, could be a third party too. But as a consumer that has very little difference for me.
If you buy something from Tiktok, you presumably visit the merchant's website, which almost surely will have chosen to have a tracking pixel that sends data to FB (Instagram). You can read a bit about how tracking pixels work here: https://jvns.ca/blog/how-tracking-pixels-work/
In this case it's not Tiktok and Instagram that are sharing data with each other, but the product website that is choosing to share data with both of them.
It's because you are the exact target demographic consumer for that product, and it's visible in your behavior patterns when using your apps combined with what they know about you (age, sex, location, demographics, etc.)
Much more likely an explicit retargeting ad network that doesn't realise they already bought it.
Retargeting has been a thing for like 15 plus years now. Visit website for knives, ad network tracking cookie notes that down, same ad network later serves you ads for the same knives. Or some convoluted data sharing network that has the same outcome these days.
It's not much more likely. People don't realize but demographic targeting works really well. When you are a certain age, certain gender, living in a certain area, with a few other inferred characteristics, you're very likely to be talking about, thinking about, and buying a small set of product types
I think the OP knows best what happened to them, so I will leave it up to them to decide. But to get targeted for a product you just looked feels like pretty standard retargetting.
I don't disagree with your point that demographics can be very good and sometimes uncanny, that aspect I agree with. It is just the timing that leads me to believe it is retargetting.