You're asking different things. In the title, you're asking if it's okay to install the same wallet on different devices. I'd say that's fine (as long as the wallet itself can be trusted).
Then, you're asking this:
Now I want to ask whether there are Wallets that are secure, synchable in between the desktop app and mobile app together?
I don't think this exists. The transactions (in a HD wallet) will of course be synced through the blockchain, but the wallet itself (including the address labels) won't get synchronized.
For example if I am going to set up the wallet on desktop then I will have to enter the Private key there. In similar ways when I will set the wallet on my phone then also I will have to set the private key there.
That will work.
Is it going to be issue to insert my private keys on two different devices?
Well, "it depends": you asked for "wallets that are secure", but you should also make sure the OS is secure. If you use the same wallet on different devices, you need to make sure they're all secure.
Will there be an issue? Only if your phone gets compromised. Or your desktop. And that's the issue: you don't know that until it's too late, but by doing so, you're doubling the risk. And that's why I wouldn't do it.
I think you need to take a step back: what are you trying to accomplish? If you own a small amount in Bitcoin that you can afford to lose, you can just do what you're suggesting. It will be very convenient and you know (and accept) the risk.
But, and I assume that applies to most people: if you want to get an increasingly larger amount of Bitcoin, it's better to create different wallets for different purposes (and that's what I do). Sometimes I use the same wallet brand, sometimes I use the same seed, sometimes it's watch-only, and some are different wallets. That doesn't mean they're all
loaded, but that's not really the point.
And then there's privacy. If you have $5 trillion in your mobile wallet, your local barista may be tempted to hit you on the head. And if you buy a jet, you may not want the seller to know you bought 2 jets from his competitor the other day.
Separate wallets for different purposes are great for privacy!