Why is that other wallet btc address could not be verified? Isn't it connected to the bitcoin network? Or only it will be connected when wallet will be used for doing new transaction which is being broadcast to the network and node will going to accept and update it as the latest transaction for bitcoin. However, I am confused with bitcoin network and that include this one signing in message though I can sign in a message in my bitcoin wallet but it could not be verified just like what it shows in the other replies.
What could be the reason behind why electrum wallets could not be verified by other signing bitcoin verification message?
It depends on each wallet. Most of wallets use BIP 39 for their mnemonic seed but Electrum has its own code to create wallet seed (not BIP 39). The difference causes troubles when you verify signature by different wallets.
If you have ever imported other wallets that support BIP 39 into Electrum, you will see the option for BIP 39.
https://electrum.readthedocs.io/en/latest/seedphrase.html