I'm not sure why you're worried about registering a name for any DEX since it's not necessary and you only do that for centralized and custodial exchanges. Are you going to declare those wallets meant for your kids in your tax returns? If that is not the case then you can simply set up a non-custodial bitcoin wallet like Electrum and give them the seed phrase once they are old enough to receive it.

When it comes to savings then you would really need to deal with non custodial ones which simply means that you are in own of the private keys of said wallets.Its up to your whether you do like to know

the keys of those wallets or only into the possession of your kids and wont tend to know for yourself for the benefit of privacy or complete ownership but since you could really
still able to monitor those transactions when it comes to savings.

One of your common problem would really be on how you would make your children to learn up everything.

Hardware wallet if you don't mind spending a few quid or a simple offline wallet will suffice. Just save some of the addresses, and you can deposit Bitcoin whenever you want without actually opening the wallet. It's probably best to educate your children on how to stay safe with Bitcoin wallets, and the most common pitfalls.

Basically, the only thing your children need is either the private key or the seed, or preferably both. So, you only need to generate them, assure that they're working properly (which you can do safely on a offline computer), and that should avoid most of the common pit falls. Don't use a online service, since the attack vector increases, you can't guarantee that they haven't seen the private key themselves, and you also can't guarantee that the online service will still exist when your children grow of age to claim the wallet.

I do advice OP to save assets in hardware wallet too rather than setting up wallet's on Dex, I assume after reading his thread that he was going to start saving in it for a long term. I love the idea of saving up a a crypto asset for our children especially if it is bitcoin and ethereum because for sure it still has a value in the future and I believe that the children will love it because I'm pretty confident at that time the adoption of crypto in their generation are widely use.  Just make sure also, that you have taken down notes all the seed phrase and keep it because it is the only way for the children to access the wallet.

I'd rather go for a software wallet (noncustodial) like electrum than buy a hardware wallet too early. Unless you think it's good enough to store hardware for years. The problem is you don't know how many years you have left to inherit those bitcoins. Also, by the time you hand HW over to your children, it may become an obsolete tool and not worth using. The durability of HW also needs to be considered.

I don't know what Decentralized Exchange (abbreviation DEX) has to do with anything here. You don't create an "account" in a DEX, if you do it is no longer a DEX. Instead you create a local wallet like any other  bitcoin wallet but it would be a hot wallet. Although it can still be used but why would you even choose it if you want to create a "wallet"? Just use any of the traditional wallets such as Electrum.

You can create a new deterministic wallet with Electrum that will give you a seed phrase which you can write down as backup, then choose a different address for each of your kids and deposit the money in there. In the end you can give them each a single private key (corresponding to their address) while still holding the seed phrase so that you can recover everything if you needed.
It is the same with hardware wallets as others suggested.

You need neither a decentralized exchange (DEX) nor a centralized exchange (CEX) in order to save money for your kids. As a name suggest, an exchange is nothing else but a place where the buyer and the seller meet to make a deal, it is not supposed to be used for storing your funds, especially if you are focused on the long term.

So, if you are focused on the future then you will likely need to set up your own bitcoin wallet. During the creation of a wallet, you will be given a set of English words that should be carefully written on paper and kept securely. These are your seed phrase, your main key. Your seed is used to generate a set of private keys. Private keys generate addresses where you send your savings.

There is also a thing called "passphrase". A passphrase is like a password, but when combined with your initial seed phrase, it generates a totally different set of private keys and therefore completely different addresses. Each passphrase generates its own set of addresses. You can utilize this feature and create multiple wallets using only one seed phrase. Each kid will have an assigned passphrase, with which they can access a particular wallet after combining it with your seed phrase.

Just get a hardware wallet? Why even bother making a wallet in an exchange when you're going to use it to hodl and earn crypto for your children. It's not like they're going to use said coins to trade right now, or in the next few years even. You're just going to let the coins sit there for a long time, till at least when your children have grown old enough to handle their own finances, then you can give them said hardware wallet without any issues whatsoever no? Trezor would be one of the top choices imo.

Why not just have Electrum wallets for each kid and send money to these wallets when you can? You can then give your kids access to their wallets when you believe they are old enough. It seems like a simple and safe solution to me. Also, I don't think it requires a name, so that part of the problem is solved automatically. After all, Bitcoin wallets don't have to be tied to particular names and identities. As for exchanges, I wouldn't trust any of them when it comes to savings. Even good ones can fail at some point over the years or, if it's a centralized exchange, change policies to those you or your kids might not find appealing in the future.

is it not too risky for you to make a wallet on dex, I think it would be better if you just use a soft wallet, which is only specifically for you to save for your child in the future, by the way how old is your child? Is your child ready to learn to operate the wallet that you will inherit?