What is the scriptural basis for a monarchical episcopate? What scriptural basis has traditionally been put forth by the Catholic and the Orthodox Churches to support the idea of monarchical episcopate (the idea that there must be only one bishop over a city or a region)?
The scriptural basis on which Catholicism support the church polity of monarchial episcopate are the following scriptural passages:
Acts 21:18

On the following day Paul went in with us to James, and all the
elders were present-(ESV)

In Catholicism, being an elder and being a bishop are two different offices.Futhermore, in Catholicism, St. James was believed to be a bishop. Therefore, Acts 21:18 is an explicit Bible verse that supports monarchial episcopate when understood in terms of overall Catholic teachings.
Titus 1:5

This is why I left you in Crete, so that you might put what
remained into order, and appoint elders in every town as I
directed you—(ESV)

Catholics believe that bishoprick and eldership aren't the same.Catholics believe that St. Titus was a bishop.In this verse, St. Titus appoints elders. In Catholicism, appointing elders is the function of a bishop.When read in Catholic lenses, Titus 1:5 is an another Bible verse that clearly supports monarchial episcopacy.
1 Timothy 5:17-22

Let the elders who rule well be considered worthy of double honor,
especially those who labor in preaching and teaching. 18 For the
Scripture says, “You shall not muzzle an ox when it treads out the
grain,” and, “The laborer deserves his wages.” 19 Do not
admit(you-singular) a charge against an elder except on the evidence
of two or three witnesses. 20 As for those who persist in sin, rebuke
(you-singular) them in the presence of all, so that the rest may stand
in fear. 21 In the presence of God and of Christ Jesus and of the
elect angels I charge you to keep these rules without prejudging,
doing nothing from partiality. 22 Do not be hasty in the laying on of
hands(you-singular), nor take part in the sins of others; keep
yourself pure -(ESV).

St. Timothy is a bishop according to Catholicism.In this text, he possesses exclusive authority over the Ephesian elders.1 Timothy 5:17-22 supports monarchial episcopacy when read in light of Catholic theology.
Conclusion
Ample scriptural passages provided cogent biblical evidence for monarchial episcopate. This is only true when read in light of Catholic interpretation of eldership and bishoprick, though.

References:
http://www.biblicalcatholic.com/apologetics/a80.htm
http://www.newadvent.org/cathen/02581b.htm