StevenS

Welcome to the Serif Affinity forums, @ThatOneSerialKey.
The most common reason for one of the applications not showing up in Downloads and Product Keys is that it was not purchased from Serif, but instead was purchased from the Microsoft Store or the Mac App Store.
In that case, you would simply need to sign in to that Store on your son's laptop, using your credentials, and you would find it there in the library of purchased programs, ready to download again. 
If that does not provide the answer for you, it will be best to contact affinity@serif.com via email and provide full information to them, as some of the needed information for further investigation should not be provided here in public for privacy reasons,

@t.lange
Welcome to the Affinity forums.
Since you purchased the app some years ago it is quite possible you bought it from the Mac App Store.  If that is the case, you will not have a "licence key" since verification is handled within your App Store account.  If this is your situation you need to ensure you are logged into your App Store account, using the same ID as you used when you purchased the app and it should be immediately verified and activated.
OTOH, if you actually bought the app directly from Serif, you need to log into your Affinity Store account to find your licence key.

Hi @Kelly Bellis, welcome to the Affinity forum.
Below is an image that shows the current free content, taken from the cart after Affinity Photo has been added to the basket on the Affinity Store.
If our apps are bought through the App Store or the Microsoft Store, the (same) free content is claimed when the app is registered.

There are several free items made available when you actually purchase the application. You can see what they are if you add the application to your cart at the Affinity Store. You can then delete the application from your cart if you don't actually want to buy right now.

The process of setting up an account (at store.serif.com) begins when you start to download a product, for example, a free trial of Affinity Photo. This is separate and apart from your forum account (at forum.affinity.serif.com)

@Donglobal
Short: Yes
 

As StevenS said above:
If you now have a different Apple ID to the one you originally purchased with then you will need to purchase a new licence of the Affinity software if you want to install it again on a different machine.
Unfortunately, there’s nothing that Serif can do about this, but you can try asking Apple if you want to go down that particular ‘rabbit hole’.

Welcome to the Serif Affinity forums.
The iPad apps are sold by Apple, from the iPad App Store, not by Serif.
Therefore, I think you'll have to contact Apple Support to get a refund.

Hi Bryan Rieger,
I can assure you this tweet doesn't align itself with our views as a company or as individuals, this was almost certainly a mistake and I'll get this rectified asap.
Thanks for bringing this to my attention.
C

However, if the question was serious: I don't think the TestFlight signup is controlled-access. It was, long ago, but is now open-access as far as I know, requiring only that you own the appropriate retail Affinity application.
When there is a beta (there is not one currently) you just click on the link in the beta announcement. For example, the link in this one, for the prior Designer on iPad beta:
(Again, the link in that one won't work as there is no beta currently.)
However, I'm sure the team would still enjoy donuts  

Welcome to the Serif Affinity forums, @Jacob Eickhardt.
There is no advertised discount for purchasing another Affinity application when you already have one. You could, though, write to serif@affinity.com and see what they say. Yes, there is a Trial version available. You can download and install it, and then you'll have 10 consecutive days from the time you first open the application. After that, if you purchase directly from Serif, you can have another 14-day period to keep using it, and anytime within that 14 days you could decide to return it for a refund.

Mac Free Trial
Windows Free Trial

Or visit https://affinity.store and click Publisher in the top navigation bar, and you'll fine the trial links at the bottom of the page.

Welcome to the Serif Affinity forums.
If you purchased from the Affinity Store, and the purchase was successful, you should be able to login at https://affinity.store/account/downloads where you will find the product name and download link, along with the product key and email address you'll need to use during the installation.

Thanks Walt  I did miss that part at the end in my haste to give OP the correct information.
Further down the email that was originally shared there will be an area like the one below that does include info concerning when the voucher code will expire. However there is some discretion and OP can reply to the email if/when they choose to get the app and we would gladly honour the discount offered.
10% OFF AFFINITY DESIGNER *DISCOUNT CODE HERE* Your voucher will automatically be applied to your basket Your voucher will expire in 7 days.

Thanks Walt  I did miss that part at the end in my haste to give OP the correct information.
Further down the email that was originally shared there will be an area like the one below that does include info concerning when the voucher code will expire. However there is some discretion and OP can reply to the email if/when they choose to get the app and we would gladly honour the discount offered.
10% OFF AFFINITY DESIGNER *DISCOUNT CODE HERE* Your voucher will automatically be applied to your basket Your voucher will expire in 7 days.

But if, upon launching, there is a button to "Activate Trial" and you exit the program without pressing that button, I would hope the trial clock wouldn't start.

Still unanswered, I think, is the question of whether they can still get the 10% discount offered in the email if they start the trial later.

It is the second one. We send it out 10 days after the user signs up, but as we advised OP - they "still have a full 10 days to play with the app", if they've not clicked the "Start My Trial" option.
Thank you for signing up to our trial, with links to tutorials etc to help people get the most from the app.
Would it be clearer if I wrote...
We send out that email 10 days after you originally signed up for the trial on the presumption you will be so excited to try it out immediately etc etc
I think the important thing is that OP understood the message and that they can still use the trial.

It is the second one. We send it out 10 days after the user signs up, but as we advised OP - they "still have a full 10 days to play with the app", if they've not clicked the "Start My Trial" option.
Thank you for signing up to our trial, with links to tutorials etc to help people get the most from the app.
Would it be clearer if I wrote...
We send out that email 10 days after you originally signed up for the trial on the presumption you will be so excited to try it out immediately etc etc
I think the important thing is that OP understood the message and that they can still use the trial.

Hi @Affinity_Pro, welcome to the Affinity forum.
If you've not yet clicked "Start My Trial" you will still have access to it.
We send out the email you received presuming that people will be so excited to try it out, that they won't be able to resist using it the moment they have access to it. 10 days later, we send the email informing that the trial has expired.
So don't worry, if you've not yet began the trial, you still have a full 10 days to play with the app and see what you think.

Welcome to the Serif Affinity forums, @pascal rumilly.
There is a problem with registration that will be fixed in 1.10.5 when it is released, which should be soon. It is already fixed in the 1.10.5 betas for Windows.
For additional information and workarounds for 1.10.4:
 

This is the case when the trial has not yet been started. 

This is the case when the trial has not yet been started. 

Hello and welcome to the forums, @Jazh
Purchases made in the Microsoft Store or the Apple Store are automatically activated. 
To use the programmes on a new PC, you must use the same account details that you used when purchasing from either store.
After logging in to the MS Store, you will find your apps in the library. 
And more to read here: 
 

Welcome to the Serif Affinity forums, @A. Rabbit.
There is no multiple-application discount advertised. However, you might try emailing affinityorders@serif.com and inquiring directly. Perhaps there is something that they don't advertise. Note that (as mentioned above) they are officially closed until January 4, so I'm not sure when they'll handle your email.