Legit looking iPhone Settings Pop-up

This does not look like a message Apple would have iOS put up. It looks like a "phishing" scam message.

I suspect that some sort of Web notification (and/or rogue app that slipped through Apple's App Store screening) is involved, and that the use of the system settings icon is unauthorized – meant to perpetuate the scam.

Maybe you have Safari open in the background, maybe there is some Safari process to process notifications even when you are not actively using Safari – but a "phishing" notification from a Web site is a much more likely explanation than this being a legitimate message from the system.

Dead giveaways:

• "The SIM card will be damaged." Frightening, but how will it be damaged? And why would this damage occur on a precise 1 minute and 3 second schedule? To me, it doesn't pass the sniff test.
• "Install the PowerfulCleaner app, open it, and run the cleaning procedure." If Apple wanted you to take action, they'd tell you to do it using the built-in tools in iOS or to download an Apple app.
• "Your device Memory is full!!" Mixed caps suggest that writing professional text is not the author's strong point. Nor do they seem to understand the difference between memory (RAM) and storage. Also, if your storage is full, exactly how are you supposed to download this "PowerfulCleaner app" that will need some of it?

As far as I'm concerned, the mention of the "PowerfulCleaner app" is a good reason for Apple to investigate this app and its developer. Maybe it's legitimate, but I don't feel like installing it on my iPhone based on the way that this message is pushing it.

If you are getting those messages while surfing on Safari, then please ignore they are spam and could be a scam too.

iOS / iPadOS devices cannot be infected with Virus / Malware / Spyware unless you have intentionally downloaded spurious software or unauthorized apps directly from the internet and installed them on your device or/and have Jailbroken. 

iOS 15 and 16 support some form of push notifications ("Safari Push Notifications" or "web push notifications"). That makes me think that Safari doesn't need to be open – that there's a server process on your phone that will listen for these notifications, once you have registered with the Web sites that send them.

Sending web push notifications in web apps, Safari, and other browsers | Apple Developer Documentation

As for the notices appearing the day after you downloaded the security update, maybe you just happened to visit some Web page that signed you up for notifications at around the same time you did the update.

To improve understanding of the phenomenon, it would be highly beneficial if you could share a screenshot. Visual representations carry greater impact than written descriptions.

Please remember that this is a public forum, so when attaching a screenshot, please avoid including any personal credentials such as IP addresses, card details, email IDs, Apple IDs, IMEI numbers, serial numbers, phone numbers, order IDs, or any identifiable location information if you are sharing a map.

With the limited information that you have provided, It is hard to even guess the problems faced by you. 

You may get faster and better responses to your query by --> Writing an effective Apple Support Commun… - Apple ...

Please re-post with detail so that someone else can help.

You are likely to get such popup messages on your device only if your device is Jailbroken.