Mail can be extended to mimic pretty closely the feature you mention from Outlook.
OS X provides a feature called services that can be used to extend the functionality of an application (see http://www.macosxautomation.com/services/learn/index.html for more information).
I will use Automator and AppleScript to create a service that moves sent messages to the folder of the message being replied to, and then assign the standard reply keyboard shortcut (⌘R) to the service:

- Select Utilities under Library and drag Run AppleScript to the empty pane on the right:

- Configure the service to receive no input in Mail:

Replace the contents of the Run AppleScript action with:
on run {input, parameters}
tell application "Mail"
try
(* Get selected messages or exit *)
set messageSelection to selection
set selectedMessage to item 1 of messageSelection
set replyMessage to reply selectedMessage opening window yes
set messageID to message id of selectedMessage
set currentMailbox to mailbox of selectedMessage
on error
return
end try
repeat
(* Wait until reply message is sent *)
delay 2
try
if replyMessage is not visible then exit repeat
on error
exit repeat
end try
end repeat
synchronize with (account of currentMailbox)
(* Wait while reply message is being sent *)
delay 10
(* Loop over all sent messages *)
set sentMailbox to sent mailbox of application "Mail"
set allSentMessages to messages of sentMailbox
repeat with sentMessage in allSentMessages
if source of sentMessage contains messageID then
set mailbox of sentMessage to currentMailbox
return
end if
end repeat
end tell
return input
end run
Save with a descriptive name like Reply and Move Sent Message to Current Folder. The service will be saved to ~/Library/Services/
. This is what the service looks like:

Open System Preferences, select the Keyboard preference pane and then the Keyboard Shortcuts tab.
Select Application Shortcuts, press the + button and change for application Mail the shortcut assigned to menu item Reply to ⌥⌘R. We change it to avoid conflicts when assigning ⌘R to the service below:

- Select Services, scroll down to the service you previously added and set the keyboard shortcut to ⌘R:

- Open Mail. Notice that Reply's shortcut is no longer ⌘R, but ⌥⌘R:

- Also notice that there is a new service, accessible with ⌘R:

- Now select a message in a folder, press ⌘R, type your reply and send it, in a few seconds you should see the sent message appear in the folder.
Changing the service to "Reply All":
Olivier has asked in a comment how to expand the Automator Service to reply to the sender and other recipients:
If you are using a recent version of macOS, choose "Quick Action" in Automator.
Replace the following line in the AppleScript above:
set replyMessage to reply selectedMessage opening window yes
with:
set replyMessage to reply selectedMessage opening window yes reply to all yes
Change the application shortcut for "Reply All" (shiftcommandR) to something like shiftoptioncommandR and bind shiftcommandR to the service.
The script has some limitations and side effects:
The service can't tell canceled and sent messages apart. So, if you press ⌘R and then change your mind and close the window, the service will continue executing and check your Sent folder for matching IDs.
AppleScript doesn't offer any means of getting a list of replies to a message. My workaround is to get the message id of the message being replied to and search it in the sent messages.
The sent message can't be moved while being downloaded. AppleScript doesn't return any error message if that's the case, so I added a 10-second delay. A 5-second delay was not reliable in my tests, adapt it to your needs if necessary.
While the service searches the Sent folder, Mail slows down.