Q: Pages License

What specific version of OS X (Apple logo : About This Mac) is on the “new” Mac, and what specific version of Pages (Pages : About Pages) are you referring too?

Apple only keeps the most current release of Pages in the OS X App Store, and this is now v6.0 which requires macOS Sierra. All releases of Pages older than v6.0 are no longer available from Apple, and cannot be downloaded again from your OS X App Store Purchased tab.

You can continue to use Pages v5.5.3 on El Capitan, and ignore the “nagging” about needing a newer version of Pages (which you cannot install on El Capitan anyway). I can tell you from my own testing, that Pages v5.5.3 will run in macOS Sierra providing that you do not attempt to open a Pages document that was created, or edited by Pages v5.6 or later.

A new (not used) Mac purchase should also permit one free copy of Pages from the OS X App Store, but due to the constraints of the first paragraph, you would have to update to macOS Sierra beforehand.

Pages v6 will allow you to perform auto-capitalization of first word of sentence (in concert with new macOS Sierra feature), have multiple opened documents in tabs, and fixes hyperlinked, ToC page numbers in exported PDF. And, it introduces document collaboration... (yawn).

when I try to open documents created with my old Mac I am prompted to update Pages to a newer version

So what version of Pages were those created with?

And, if your old applications were ported over with Time Machine, please look in your Applications Folder. I have both an older version (4.3) in a folder "iWork" and the latest version (named simply "Pages") in that folder. By default, the OS will open (or try to open) any Pages document with the latest version. Note: once you open it with the latest version and edit it, you cannot go backwards with it. If you need to use the older version (if you have it), simply highlight one of your documents' icons, press the Command + I keys to see a get info window. Scroll down to "Open with" and choose Pages 4.3 (you'll see that the latest version is the default). If you want to use it with all your documents, choose "change all".

FWIW, I use the older version (I prefer it).

Open the Mac App store and check for updates. Update Pages to the current version from there. It is not part of macOS and is not automagically updated when you update macOS.

That is a new one. So you're sure that you only have the 5.5.3 version on your new iMac?

And, have you checked your Purchases section in the app store to see if the newer version is available for download?

And you did check your Applications folder - both under "iWork" folder and a separate listing for Pages app?

You said that you purchased a new computer? Pages, etc. is included when you purchase a new one, but you must "accept" it in your Purchases section in the app store. If it does not, you need to address that with Apple - you have a warranty and 90 days of phone support, so I'd suggest you either call them or visit your nearest Genius Bar to get this sorted out. In case you purchased yours from an authorized reseller, make sure you have your receipt as proof.

If the Mac is "new to you", but purchased used, then those apps are not included and only available if you purchase them.

Just to reiterate, even with OS X 10.11.6 on your Mac, you are marooned at v5.5.3, because the only version of Pages available from the App Store is now v6 — which requires macOS Sierra. Your Purchased tab Pages entry is now pointing to v6, not any prior Pages download.

Rename your Applications : Pages (v5.5.3) to Applications : Pagesv5.5.3, and then bring that Applications : Pages (v5.6.2) on your older Mac over and place it in the new Macs Applications folder. Now, you can open documents created by v5.6.2.

You rename first, in case your copy of Pages v5.6.2 from another machine is damaged in the copying process. Once you are satisfied that Pages v5.6.2 copied from your older Mac works as expected, then you can backup/remove Pages v5.5.3.

Its just CYA.