That "wallet thing" is actually a storage of private keys that has the capability to spend coins from the blockchain which is stored in each of the full nodes.
The security is up to your capability (and the wallet software of course).

"Other wallets" like Coinbase aren't in the list because those are not actually wallets but a custodial service where that company is the actually hodler of your coins.
This time, they are the ones that will protect your coins and all you have to keep safe is your account and password.

There's already a list in the OP that have entries for both Android, IOS and Desktop.
Based from what I've said above about custodial wallets, as long as both wallets contain the same private keys, they'll work simultaneously without issues even if it's not the same wallet.
You can achieve that by restoring the wallet's backup (usually a 12/24-word phrase) to the other, but some aren't compatible.

Note: Given that you want to use it in both platforms and will contain the same keys, if one was compromised, the other will be as well.

TL;DR
One simple answer: By looking at your other post, wallets do not contain free bitcoins even in the early days of Bitcoin,
most wallets are free to use and I don't think there are any "pay-to-use wallets".
Do an extensive research about "custodial wallets" and after you gained confidence, you can now send it to that wallet.
Here's the board for wallet software where you can ask questions: https://bitcointalk.org/index.php?board=37.0