You do not need to own the game on Steam or use a cross save.

If you have a Steam account ,you can get this as a DLC even if you do not own the game but the problem or annoyance is that you have to link the GOG version to Steam to claim this.

Since this appears to be a re-colorization of an existing ship you might be able to find the seeds for this and add it using the save editor. For that matter, it might also be that this can or will be added directly to the editor to claim directly.

Could you provide instructions on how to do these 2 suggestions?

Sign into your Steam account
Navigate to the Steam Store
Click on the article about Starborn Phoenix Ship
Scroll down the article to find the line "Claim your Starborn Phoenix starship on Steam here."
Steam users have complained that if you try to add this directly from a link in this article that it will not be added. Instead, you need to click on the hyperlink at the word "here."
There should be a link to install DLC or something similar. I completed this so have a different message.

You should be able to see a message or note that this has been added to your library.

I have not linked No Man's Sky to Steam as I bought it from GOG but somewhere in the Steam client there will be a link to link games from other sellers and do that for No Man's Sky. Then the DLC should be available through the Quicksilver Vendor as a 4th option and you probably have to pay nanites based upon difficulty setting to add the Starborn ship.

I claimed the ship just in case but I have little to no desire to play No Man's Sky through Steam so am not linking non-Steam games to Steam. For that matter, I had the pre-order DLC starship from Hello Games but have never downloaded it. The original starship from my day one save worked out just fine and had the Protonix Core so I was good. I am still playing that save except for expeditions.

You only have until December 19th to add the DLC for the ship and you have to account for time zone. The time limit only applies to the DLC of the ship and not to buying the ship in-game which should be available across all saves within the game.

Good Luck

It can be obtained via the Goatfungus save editor by importing the ship data from a Steam save, i first tried editing the GoG NMS account data to match the Steam version but had no joy, but discovered that just exporting the ship data and then importing into the GoG save worked, link to ship data below

Simply load your save into the editor, open ship tab and click import, select ship file, once info is in the editor make a save and then load game and check to see if it's there in the summon list

This should also work for any NMS save file that can be loaded into the editor

Edit: Make sure that you have a free ship slot before adding data to the editor

https://drive.google.com/file/d/1vZ8hocQROuBAGMIVitS7zdRpFmn3hJt9/view?usp=sharing

If link does not work let me know, pretty sure i've got it set so it can be viewed by everyone with the link

This does work, thank you! I had an error when I first tried it, but that was because all of my ship slots were full. I deleted one, and was able to import the Phoenix!

sadly it didn't work for me ,also gog galaxy tries but fails to connect my steam account

does anyone know what date this was from ? as i'm pretty sure i had collected this but it's not showing up in my collection saying something to the degree of complete this expedition to claim

Do you have the Game Editor from Goatfungus? That's required to use the file. It's available on Github. Once installed, run it and open your saved game. Go to the ships tab and make sure you have an available slot to add a ship. Click Import and select the downloaded file. This will add the ship. If it fails, it may mean you don't have an available slot for a new ship. You'll have to delete one. You can export a ship to enable it to be loaded again later. Once that's done, save the edit and run NMS.

The only way the ship file in the link works is to use the Save Editor. Nothing else will do it in this way.