"mun" can stand for "minun" in spoken Finnish. The sentence decides its meaning.
Spoken form of 'minun': the -ni ending of 'työkaverini' drops
Kato, toi on mun kaveri. - Look, that's my friend.
mun here: my - spoken form of 'minun'
Written Finnish: Katso, tuo on minun kaverini.
Tää on mun äiti. - This is my mom.
mun here: my - spoken 'minun', and no -ni ending on the noun
Written Finnish: Tämä on minun äitini.
Mun isä on viel töissä. - My dad is still at work.
mun here: my - spoken 'minun'
Written Finnish: Minun isäni on vielä töissä.
Toi on mun paras kaveri. - That's my best friend.
mun here: my - spoken 'minun'
Written Finnish: Tuo on minun paras kaverini.
Mun mummo soittaa mulle joka päivä. - My grandma calls me every day.
mun here: my - spoken 'minun'
Written Finnish: Minun isoäitini soittaa minulle joka päivä.
Mun kurkku on tosi kipee. - My throat is really sore.
mun here: my - spoken 'minun'
Written Finnish: Minun kurkkuni on todella kipeä.
Spoken and written Finnish are different registers. The examples show how "mun" is used here and pair it with the written sentence. Learn the meaning in context rather than replacing every occurrence of a word.