Using "owner = msg.sender" in a non-constructor function or in a declaration statement?

Based on the clarification in the comments to smarx's answer.

It depends what you mean by "problem". Consider this function.

function changeOwner() public {
  owner = msg.sender;
}

That allows anyone to seize the owner role, assuming owner is pivotal to access control somewhere else. That's not a problem in the sense that it works. That is a problem in the sense that anyone can do it and that's usually not desirable.

Consider this variant:

function changeOwner(address newOwner) public {
  require(msg.sender = owner);
  owner = newOwner;
}

The require() strictly limits this to the owner, so the sender has to be the owner in order to relinquish ownership. It doesn't make sense to relinquish to itself, so another address (the newOwner) is passed in. A production-ready implementation of this common pattern would check that the new address is valid to prevent accidents.

It may be helpful to remember that owner is not special in any way. It's merely a variable of type address given meaning by other aspects of the contract such as conditionals that oversee permission.

Hope it helps.

owner = msg.sender; statement in a non-constructor function?

If you write owner = msg.sender; in a non-constructor function then 1) You have to pay more gas amount(approx 1.5x more) as compared to we deploy this statement using the constructor.

2) You need to call a function to set owner = msg.sender when you are using the non-constructor function.

What if instead of writing the above code in a function, we initialize the owner at the time of declaration like this in a state variable:

There is no problem with initializing owner at the time of declaration but it changes the meaning of state variable for solidity compiler. let me explain you:

Private:

When you declare owner as a private state variable like this address private owner = msg.sender; then solidity compiler treat owner variable as variable and for getting this variable value you need to define another function like getOwnerAddress().

Public:

If you set owner as public like this address public owner = msg.sender; then solidity compiler treat owner variable as functionowner() and this will also return the address value of owner there is no need to define a separate function for getting value.

For more understanding, you can see a difference in abi of contract details.

Statements aren't allowed outside of functions, but initializations are.

So your second snippet (address private owner = msg.sender;) is good.