The Slovakia centre-back is running down the final six months of his deal with the Serie A giants and will be joining PSG on a free transfer at the end of the season.
The Ligue 1 champions tried and failed to bring Skriniar to the club last summer, with Inter holding out for an extortionate fee amid the hope of tying the 27-year-old down to fresh terms.
However, Skriniar has rejected the opportunity to extend his stay at San Siro and will now leave the club for nothing in the summer, with Inter also rejecting the chance to recoup a fee for him in January.
PSG reportedly tried to accelerate Skriniar's move to the Parc des Princes during the January transfer window, but they failed to fork out the €20m (£18m) fee that Inter were demanding.
The former Sampdoria man will therefore see out the remainder of the campaign with Inter, but he will not be wearing the captain's armband during his final few months at the club.
Ahead of Sunday's Milan derby with Scudetto holders Inter Milan, Inzaghi told a press conference: "I don't judge his choices, but the man and the player, who are unique.
"He is a splendid boy who works well every day. He will most likely be there from the game. Captain's armband? There are hierarchies. Due to what happened, Skriniar will no longer be captain, but we have already talked about it serenely with him."
Skriniar had worn the armband for 10 consecutive Serie A games prior to last weekend's win over Cremonese, where he sat out through suspension after seeing red in Inter's 1-0 loss to Empoli on January 23.
With the centre-back stripped of the captaincy, Inzaghi has revealed that goalkeeper Samir Handanovic will assume skipper responsibilities, while Marcelo Brozovic, Danilo D'Ambrosio and Lautaro Martinez are also part of the leadership group.
Amid rumours that Skriniar will be booed by Inter supporters at San Siro, Inzaghi affirmed that he had "no message" for the Nerazzurri faithful ahead of a possible frosty reception.
"Our captain is Handanovic, then it is appearance-based: there are Brozovic, D' Ambrosio, Lautaro. It will change from time to time," the Inter boss added.
"The fans' reception for him? I don't have any message to give to the fans, I see him working every day and he works great. These things in football are now agendas and you have to get used to it."
Skriniar has 11 goals and five assists to his name from 242 games in an Inter Milan shirt since arriving from Sampdoria in 2017, playing an integral role under Antonio Conte as the Nerazzurri won the 2020-21 Serie A title.
The Slovakian has started 18 Serie A games for Inter this term despite his contract situation, and he will be expected to return to the first XI in Sunday's Derby della Madonnina.