"I am a civil servant."
"kedi" ends in a vowel and so needs the "-y-" then "-im". "memur ends in a consonant and just gets "-um". https://www.duolingo.com/comment/7738932 Turkish Grammar Portal https://www.duolingo.com/comment/7738500 To Be: Copula https://www.duolingo.com/comment/8612302 Possessive Suffixes https://www.duolingo.com/comment/9041808 4-way Vowel Harmony
You are right that the form in first person also looks like the possessive, but you would put "benim memurum for "my civil servant" and "Ben memurum." will always be understood as "I am a civil servant."
"memur" means "civil servant", the suffix "um" is to indicate the form of the verb "to be" that is conjugated for "ben" or "I" as in "I am". With "benim" the possessive suffix also comes out as "um" for the first person, please understand that the "u" is the vowel because the last vowel in the word is "u" or "o" for both of the suffixes and that is governed by "4-way vowel harmony". Note that if you wanted to say "You are my civil servant.", the possessive suffix for "my" is added first and then the verb form for "you", so that would be "Sen benim memurumsun." If I wanted to say "I am your civil servant.", then that would be "Ben senin memurunum." with the possessive suffix for "your" first and then the verb form for "I". I had the wrong link for the possessive explanation so I have fixed that. Here is a dictionary for you: http://www.turkishdictionary.net/?word=memur but that doesn't help with the suffixes so there is a second dictionary for those: http://www.dnathan.com/language/turkish/tsd/
If you wanted to say "I am my own civil servant.", note that we add "own" in English to make that work and in Turkish the sentence looks different also. "Ben kendi ...."? Someone else showed it in a different discussion and I cannot remember exactly how it looked, but it was different than I expected.