The sentence structure is a bit unusual, like this:
Is this proper use of the sentence structure? Shouldn't it be "Me am" instead? Considering sentences starting with "I" are in the form of "I am...".
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What you are referring to is a bit odd since it is not really "English", per se. As tunny states, you cannot use "me" as the subject of a verb.
However, the line of chat text you have quoted is actually:
/me is tired this morning
Note the leading slash. This is not meant to be proper English, the "/me" is a common command in IRC chat rooms to represent an action, and is often used online by people familiar with that command outside that context to represent the same thing, with the expectation that it will be recognized as such (and sometimes, it's not). In IRC, the /me would actually expand to the user's name, perhaps with a special color or indicator to indicate an "action". So the above would have been seen by others as something like:
Frank is tired this morning
As SE chat is not IRC, it does not recognize this as a command. It can be confusing to look at if you read it as English but really it is a form of chat-room "slang" for representing an action, and the original intent of "/me" was for interpretation by a computer, not a human.
'Me' in this respect is considered ungrammatical by purist although it's most common in colloquial English. You of course know the joke: St. Peter (who has the key to heaven) asks the question: 'Who is the next?' Somebody replies: 'I am.' St. Peter: 'Teachers of English, you just stay at the end of the queue.'