You can keep both. They are seperate pieces of armor. They simply can’t be combined into one.
While “enchanting” does exist within DnD it is something that is very rarely seen in Homegames, and even more rarely in Oneshots. The reason as to why is because it is canonically a very involed, time and ressource intensive process. I’ll quote from the Downtime Revisited section of Xanathar’s Guide:
“Creating a magic item requires more than just time, effort, and materials. It is a longterm process that involves one or more adventures to track down rare materials and the lore needed to create the item.”
While possible, this process is usually not played out during one shots and if it is, you will know because it’ll be a main story beat in the Oneshot.
So, just for the future: If a DM writes down a list of magic items they are either going to be items you found/looted during the adventure, or they are going to be items which your questgiver might give you as a reward for a job well done. So they are always additional items you receive and never to be understood as improvements for your existing items.
“Enchanting services” would certainly be under the mantle of Homebrew Story Awards and if the resulting item were to combine the magical properties of multiple magic items (e.g.: Cast off armor + +1 armor) it would certainly be a homebrew item, as such an item (usually - see Tankanaris example below) does not exist in the DnD canon.
Last but not least: If a DM doesn’t specify the type of magic item (usually happens with weapons or armor) then the implication is that you may choose your own within the limits of the magic item description.
For example:
While regular +1 armor is possible with any armor type (This link gives you the full list while this source doesn’t but does specify that light, medium and heavy armor are eligiable) … Adamantine Armor specifies that the eligable armor has be either medium or heavy (excluding hide armor).
I hope this clears some things up for you guys 