I have one blizzard account but two WoW accounts one is on the EU server and another on a US server. I am currently playing on my US WoW account, but I like to know if my son can play on my EU account at the same time, I am currently stuck overseas and my son is on lockdown at the house.
Will this be a problem?
Yes. Your son can’t play on your EU account, at all. Account sharing is prohibited.
You need to get him his own account.
Depends how old your son is.
Actually they can, if they are 16 or under and you are their guardian.
Unfortunately yes because you can’t access a battle.net account from two different PC’s. You will be forever kicking each other off the account which is probably not the experience you would want.
Best thing to do would be to have your son have his own Blizzard account so that he can play on it whenever he likes. If you are going down that route though, do make sure to keep it secured with the free Blizzard Authenticator.
Best of luck now for all your family in the adventures to come 
He can login to the game without using battle.net application!
Not recommended!! When playing Blizzard games, always use the desktop application to login and launch the game.
Sorry I misread the first post.
You are not an MvP and you dont know the rules.
If the child is under 16 they can use it .
Many times i have seen you given wrong info please stop .
But it can be done and has to be done at times when battlenet is down which it has been alot in BFA and recently.
There is also an option the exit the app completely when you run the game. If they choose this then it can be opened somewhere else without kicking them off.
For sure, it can be done however it wouldn’t be something we would recommend doing. On a case by case basis like you highlighted it may offer you a way to login to the game but it still would not be something Blizzard CS recommends as a way to connect to the game.
As also highlighted by Dottie, there is another way around if you want to keep the same setup but all I can say is that this method is not recommended. If you want to play together at the same time, I can only encourage you both to have your own separate battle.net accounts and if needed you can setup the appropriate parental controls on the child’s account.
If the OP or anyone else reading this in the future needs help with any form of account setup, feel free to contact our CS team
Thank you very much for your reply 
Are you a member of Blizzards staff? I don’t believe you are, therefore you have no right to tell anyone they can’t post. If you are, then just suspend my forum account, and move on, then I can make a formal complaint about you abusing your privilege for personal reasons.
You are giving incorrect information though.