Christian democrats almost always point to 1 Samuel 8 to criticize monarchy. This is because it is the only passage in the bible that could be used to criticize monarchy, every other passage that touches on the subject always speaks of monarchy positivley.
If one does an careful exegesis of 1 Samuel 8, one will discover it is actually an endorsement of monarchy, not an attack. God is not criticizing Israel for wanting a king, but a king “like all the nations [that] may judge us and go out before us and fight our battles” (1 Sam 8:20). What the Israelites want here is an idol. “Judging” and “fighting our battles” are things that God does (for the latter, see the book of Joshua). They want a king that does what God does, so they want a god. This is why God describes their desire as them having “rejected Me, that I should not reign over them” (1 Sam 8:7). The Israelites insist on wanting a king, and God describes the king as one that “you have chosen for yourselves” (1 Sam 8:18). Saul is a populist leader. He is exactly what the people think they want, but he ultimatley ends up taking from the people and so they will regret their choice (1 Sam 8:11-18). Democracy naturally produces populist leaders. The people will always desire to have a king over themself, and democracy naturally produces that.
David in contrast is a philosopher king. He seeks after God, and writes many psalms. Solomon too is a philosopher king as he seeks after wisdom. While neither of these kings is perfect, they both desire God. A true king does not desire to rule over the people to be a god in himself, but desires to be a servant of God. When one is a servant, they are exalted. When one exalts themself, they are brought down. This is ultimatley the message of 1 Samuel 8.
It would also be quite strange if God criticized the Israelites for wanting a king given that he had promised that he would bring one (Genesis 17:15, Genesis 35:11, Genesis 49:10, Deut. 17:14-15). The book of judges also explicitly diagnoses the problem in Israel as the fact that there was no king (Judges 21:25).
“Fear the Lord and the king, my son, and do not join with rebellious officials,” – Proverbs 24:21
“Honor all people. Love the brotherhood. Fear God. Honor the king.” – 1 Peter 2:17
The three most ancient opinions about God are atheism (or anarchy), polytheism (or polyarchy), and monotheism (or monarchy). The children of Greece played with the first two; let us leave them to their games. For anarchy is disorder: and polyarchy implies factious division, and therefore anarchy and disorder. Both these lead in the same direction – to disorder; and disorder leads to disintegration; for disorder is the prelude to disintegration. What we honour is monarchy. – St. Gregory the Theologian
“God has counted the Emperors worthy to rule over His inheritance, over His earthly Church.” – St. Gregory Palamas
“In hell there is democracy and in Heaven there is kingdom.” – St. John of Kronstadt