A highly underrated feature of LibreOffice Draw is the ability to edit PDFs. Sure it might mess up a font here or there, but there's no way in hell I'm going to pay adobe an absurd amount of money for that one feature.
Are you thinking of OpenOffice, maybe? OnlyOffice's latest stable release is only three weeks ago, while OpenOffice has only had incremental updates to version 4.1 since 2014.
Looks like whoever you bought your laptop from installed a pirated or non registered legit copy. Short of buying a real license, there is no legal way to get rid of the banner that I am aware of.
I believe it’s common practice for Windows laptops to come preinstalled with Office 365 now. I don’t think it’s pirated. It’s just not licensed like you said.
OP, I’d recommend just using LibreOffice. It works fine for most situations.
The Office that comes pre-installed is often just a trial, and you need to buy a license or a Microsoft 365 subscription to use it. LibreOffice is a good free alternative, if you don't specifically need the Microsoft applications, and it will read and write MS Office files.
My 2nd hand laptop seller had the decency to crack the preloaded software before selling it to me. His remote access tool wasn't too resource-intense either.
A quick search seems to indicate that this is "normal" even on legit Office installations. You can check out the link below or go through these steps that the site suggests to see if it helps:
Remove Get Genuine Office by Disabling Connected Experience in Office
Open any one of Office Product, Word, Excel, PowerPoint, etc.
Click File on the top left and select Options at the end of the left menu.
A new windows will pop up. In the default General tab, find and click the Privacy Settings button.
In the Privacy Settings window, scroll down to find Enable connected experience and uncheck it.
Click OK to save changes.
(In Outlook 2016 and later, you should navigate to File > Office Account > Account Privacy > Manage Settings. Turn off this connected experience option will disable experiences that analyze your content, download online content, as well as other connected experiences like co-authoring and online file storage.)
Beware that some of them are not legit at all. They often are parts of volume licenses, which, theoretical, are linked to one company. Because of that they mostly are single use and cannot be used to activate office a second time.
If you're an invested user of Office, just get the Microsoft 365 suite. It's a decent value. If you aren't that picky, online office is pretty decent too. And of course there's always LibreOffice.
I don't think it is a trial version. A trial would not give a warning about being counterfeit. If OP legitimately paid for a laptop with Office, I think they should take it back.