In "take up your cross daily" what does "cross" refer to? Then he said to them all: “Whoever wants to be my disciple must deny themselves and take up their cross daily and follow me. (Luke 9:23)


What does Jesus mean here? What kind of cross is he referring to?
There are at least two questions here, probably three. What would his hearers have understood by it before and after his arrest and crucifixion? And again after his resurrection? Jesus spoke in Aramaic, translated into Greek by Luke and later into English. To complicate things further, the word he used for cross was probably the Latin furca, a fork. This was a term used in the Roman Army for a backpack, and as slang for a punishment yoke or gallows, not the cross used in crucifixion as Romans were mainly exempt. General Marius devised cross-shaped backpacks for his men, capable of carrying large loads, reducing the need for baggage trains. This resulted in the men being nicknamed Marius's Mules. Each day they would take up their packs and march.
You asked what Jesus would have meant by it. That will no doubt be answered fully by other contributors. Comments by him and others point to literal and non-literal answers. He knew what he was heading towards. Outside Roman slang there was no reason not to extend the use of furca for a crucifixion cross or a burden carried to that end. Non-Romans probably did.