As far as I can tell Minion AI doesn't include random elements, nor does their spawning include random elements, but minions don't always target the same enemy minions upon reaching lane. Why not? Are there random elements that aren't obvious (maybe subtle variations in spawn locations or something)?
This thread may be an interesting read for you.
What I will say is that the first wave seems to act the same each game in cases where both bot lanes give the jungler smiteless leashes. The linked thread explains why this is the case, too.
But after that, there are so many variables in play that there are going to be differences with each game. Presence of a champion could adjust how the minions acquire targets and how they move. Sometimes you push faster than others. Even killing a different minion in a given wave first could change how the rest of that wave plays out. There's also small bonus to minions on the team with the highest average level across all players.
Riot discussed this a little while back and it hasn't been fixed. Red side minions spawn something like .3 seconds before blue side after the third wave so if you watch a game without champs Red side will always push in Blue side
