How do Lazav, Dimir Mastermind and Vigor work together in Magic the Gathering?
Nope, when an ability on an object refers to itself by name it means "this object."


  201.4b If an ability of an object refers to that object by name, and an object with a different name gains that ability, each instance of
  the first name in the gained ability that refers to the first object
  by name should be treated as the second name. 
  
  Example: Quicksilver
  Elemental says, in part, “{U}: Quicksilver Elemental gains all
  activated abilities of target creature until end of turn.” If it gains
  an ability that says “{G}: Regenerate Cudgel Troll,” activating that
  ability will regenerate Quicksilver Elemental, not the Cudgel Troll it
  gained the ability from. 
  
  Example: Glacial Ray is an instant with
  “splice onto Arcane” that says “Glacial Ray deals 2 damage to target
  creature or player.” If it’s spliced onto a Kodama’s Reach, that
  Kodama’s Reach deals 2 damage to the target creature or player.
  
  Example: Dimir Doppelganger says “{1}{U}{B}: Exile target creature
  card from a graveyard. Dimir Doppelganger becomes a copy of that card
  and gains this ability.” Dimir Doppelganger’s ability is activated
  targeting a Runeclaw Bear card. The Doppelganger becomes a copy of
  Runeclaw Bear and gains an ability that should be treated as saying
  “{1}{U}{B}: Exile target creature card from a graveyard. Runeclaw Bear
  becomes a copy of that card and gains this ability.”


If the name were to be used literally, the text would read "a creature you control not named Vigor" instead of "a creature you control other than Vigor".