Omega Warranty Card Without a Serial Number: Is It Normal?

Some Omega's come with blank warranty cards and they are filled in with permanent style pen by the dealer when they get them, or at time of sale. Your watch came from an AD originally and The Luxury Well says their watches come with manufacturers warranty so your warranty looks valid. To be kosher you should fill in the Ref number, serial number and the date you received it. Then if anything goes wrong you are covered.

I'm not sure what that code number means ....

That would worry me. I just checked the Warranty Card for the new Speedmaster I bought recently and the watch number is clearly printed on, with the reference number and dealer's name, with the date inscribed on a special panel - as you would expect.
Th lack of them on your card has to mean something - is it perhaps a grey market watch? I would take it back.

It's not a huge red flag, but of course it's a grey market watch once it's out of the AD channel. I've purchased several Omegas from ADs in the last couple of years.....a CK2998 in Aruba and a ceramic Moonphase Speedmaster from a well known AD in Dallas, BOTH had the reference number and serial number hand written on the red card at time of sale, along with the date. The stores name was pre-printed on the warranty card just as the OP's pic showed. I've also had Omegas where the reference number and serial number was preprinted. Maybe Omega is changing their methods, it does seem a bit odd to issue a warranty card that is blank, to be filled in by the dealer, or someone else, sometime down the road. But Omega has issued blank warranty cards as standard procedure.

What the OP doesn't know is did the watch originally come from Govberg, or even the USA? That pictogram card looks older and doesn't match the the blank warranty card in color or current style. Looks like a mismatch. And I notice on the Pictogram card that the Operating Instruction edition to be used is: 15. We are up to manual edition 22 now, so this would be a pretty old card. If it belongs with this watch it would make the watch quite old. Just something to consider. This is the problem with grey market watches.....mismatched and cobbled together documentation.

Logical thought, but why does my Speedmaster bought from a Dallas AD in March 2018, with their store preprinted on the warranty card, have the code 20150?

You mean the brown cardboard box? You don't know if that box is the one for that watch so I wouldn't conclude that tells anything about the age of the watch. If the watch is in good condition, works properly, and you got it at a good price, and it is what you want, just enjoy it.