It doesn't have to be a full closet. It can be a dedicated shelf in a closet or cupboard, or a storage trunk, or a box under the bed. Just a place where you stockpile gifts as you buy, receive free, or make.
My gift closet is a small section of my bedroom closet. In it I have items suitable for birthday, Christmas, Valentine's, housewarming, weddings and bridal showers, and group gift exchanges. At Christmas, I raid the gift closet for stocking stuffers for my kids. And at Easter, I do the same for basket fillers. In my gift closet, I also keep components of a gift, such as jars that I plan on filling with something yummy, to use as a gift, or containers that I'll use to nicely package a hand-crafted gift.
When I have specific items for specific people, I put those items into plastic shopping bags, with handles, and hang them from a hook on the inside of the door to that closet. Just a way to insure I don't accidentally gift something intended for one person, to another, or for me to check my bag for a specific person and see if I'm about done and ready to wrap and box for shipping.
Which also brings me to this, I collect shipping boxes and mailers, year round. Mostly I'm looking for small and lightweight boxes or mailers, trying to match the size of box as closely as possible as the gift, itself. And in some instances, when I have all the components to one person's gift, I wrap, pack and label the box, ready to drop at the post office near the occasion.
I often shop, long in advance of an occasion, for specific people. When I find a great item on clearance, for someone in particular, I pick it up, even if I won't be gifting it for close to a year. Last year after Christmas, I picked up several Christmas gifts for this year. Those items are in the gift closet.
I also have several people on my gift list with spring birthdays. I shop for their gifts in the after Christmas clearance sales, too. It's a real convenience to have their gifts already bought and waiting for the actual birthday. The bonus is that I make fewer trips to stores to buy gifts. I buy for several birthdays at a time. And then I don't need to even think about it until I'm wrapping the gift.
Sometimes, this backfires on me. Like the sweater that I bought for my dad, planning on gifting it to him the next Christmas. But he passed away the June before. It still sits in a box in the closet. If I don't give it to someone this next year, I'll donate it to our church's giving tree (you find an unnamed person -- someone in need that year -- on the tree branch, with info like male/female/what they need/want, then buy their gift, and a group from our church delivers all of the gifts). But most of the time, when I find I no longer need a gift that I've pre-bought, I simply choose to gift it to someone else.
Maybe this all sounds too planned, and takes the fun out of gift shopping. But, for me, it helps me to just relax and enjoy special events and occasions.