Why do we wish "peace" to each other during Mass?

Jesus didn't tell the apostles to do it

Are you sure about that? The text in the Mass is literally a quote from Christ in the Gospel of John:

Peace I leave with you, my peace I give unto you: not as the world giveth, do I give unto you. Let not your heart be troubled, nor let it be afraid John 14:27

Thus, the idea of giving peace in the Liturgy is tied into the Gospel directly. While the ancient Church often did a kiss of peace, throughout much of the middle ages into the 20th century, the "kiss of peace" was limited to the clergy at solemn liturgies (the kiss being more of a bow bringing cheek towards cheek).

The liturgical reform in the 1960s attempted to restore some form of the ancient custom which included laypeople but adapted to the cultures of today; hence, in the United States people exchange handshakes as a sign of peace. This development has not been without critique. Some are concerned that the exchange is too unruly or disruptive. Others including Pope Benedict XVI had concerns that it would be better done at a different part of the liturgy to not distract from the Eucharist.

Cekada, Work of Human Hands: A Theological Critique of the Mass of Paul VI p. 359:

At Solemn High Mass in the traditional rite, the Pax follows this prayer. The priest kisses the altar, gives the Pax to the deacon, and it is then passed along in descending hierarchical order. When a layman receives the Pax, he does so after the clergy, who bring it to him with a pax brede (a disk inscribed with a religious symbol) that the layman kisses. In ancient times, the recep­tion of the Pax by members of laity took place only when they were separated according to sex.²⁰

Priest (principal celebrant) gives Pax to deacon:

Deacon gives Pax to sub-deacon:

Sub-deacon gives Pax to MC:

MC gives Pax to other servers:

Ibid.:

Contrary to the practice in the traditional rite, the Sign of Peace in the New Mass does not descend hierarchically from Christ (symbolized by the altar), to the priest (His representative), to the lower clergy, and to laymen (if any); rather, the clergy and laymen give the Pax to each other.

For more info comparing the post-Vatican II Novus Ordo Missæ (New Order of the Mass, "Ordinary Form") to the pre-Vatican II Catholic Mass, see the FIUV position paper #19 "The Kiss of Peace".

This article has a little background on the practice:

https://catholicphilly.com/2019/01/catholic-spirituality/beyond-a-handshake-the-profound-nature-of-the-sign-of-peace/#:~:text=ministry%20and%20message.-,The%20sign%20of%20peace%20has%20its%20liturgical%20and%20spiritual%20roots,come%20and%20offer%20your%20gift.”

An excerpt:

The sign of peace has its liturgical and spiritual roots in Jesus’ teaching on forgiveness prior to offering gifts on the altar (Mt 5:23-24). That is, if someone had a grievance against his brother, he should first “be reconciled to your brother, and then come and offer your gift.”

As Christian worship evolved, the gesture of a “kiss of peace” was a way to unite the community gathered in love and reconciliation. Without being at peace, the community and its members could not fully reflect or live the peace and love of Christ in Eucharist.

The “kiss of peace” took on various forms as liturgy developed. Sometimes it involved clergy and congregation, and sometimes only clergy. Today, we know this part of the Mass, placed just after the Lord’s Prayer and a brief invitation by the priest (“Let us offer each other the sign of peace.”), as the sign of peace.

Although the invitation is the same for each Mass, the way that peace is expressed can vary from parish to parish or diocese to diocese. Conferences of bishops or local dioceses are able to set parameters for their faith communities. Cultural and congregational considerations can play a role in what gestures best express the fullness of this gift-giving of peace.

However, according to the 2014 circular letter on “The Ritual Expression of the Gift of Peace at Mass,” released by the Vatican’s Congregation for Divine Worship and the Sacraments, actions at the sign of peace should maintain the “sacred sense of the eucharistic celebration,” and so avoid a “song for peace,” “movement of the faithful from their places to exchange the sign of peace among themselves,” “the departure of the priest from the altar in order to give the sign of peace to some of the faithful,” and should not be used as a time to express “congratulations, best wishes or condolences among those present.”

There is some latitude in the case of a funeral, where the priest may offer peace to a “small number of the faithful near the sanctuary” (General Instruction of the Roman Missal, No. 154).

In addition to the Biblical justification given, one might also consider the following:

• The beatitudes pronounce a blessing on peacemakers
• The apostle Paul used the word peace more than any other Bible writer. Peace with God and peace between people in the church was a central concern of Paul. The ministry of reconciliation is one of the church’s core missions.