There is no scriptural parallel to this command in the Torah, nor any other useful comparison where three or four or five of a commodity was required or commanded as restitution. Barnes' notes makes the following useful speculation:
The theft of an ox appears to have been regarded as a greater crime
than the theft of a sheep, because it showed a stronger purpose in
wickedness to take the larger and more powerful animal. It may have
been on similar moral ground that the thief, when he had proved his
persistency in crime by adding to his theft the slaughter, or sale, of
the animal, was to restore four times its value in the case of a sheep
(compare the marginal references), and five times its value in the
case of an ox; but if the animal was still in his possession alive
(see Exodus 22:4) he had to make only twofold restitution.
Barnes' approach seems to focus the crime on the criminal, and assumes that the penalty is intended to satisfy the severity of the criminal intent, similar to modern Western law systems. Ancient law systems tended to focus on the victim or indeed the community when setting the price for crimes, and so it's also worth considering whether this could be the more significant factor, based on other ancient parallels.
Repaying a debt of honour to the victim?
The anthropologist David Graeber studied the topic of 'social debt', where the actions of one member of the community incurred a 'debt' which must be paid to satisfy their disruption to the community. In chapter seven of Debt: the first 5000 years he explores how ancient systems sometimes bound criminals to pay back what they had stolen, and then also pay back a surplus amount to restore the dignity of the injured party. In particular he examines well documented cases from medieval Ireland (p173):
The honor price of a wealthy peasant was two and a half cows, of a
minor lord, that, plus half a cow additionally for each of his free
dependents-and since a lord, to remain a lord, had to have at least
five of these, that brought him up to at least five cows total.
Honor price is not to be confused with the actual price of a
man or woman's life. If one killed a man, one paid goods to the value
of seven [slave girls], in recompense for killing him, to which one then
added his honor price, for having offended against his dignity (by
killing him).
It's conceivable that (in a more informal manner) the extra overpayment was to restore the honour and dignity of the afflicted party, in a quantity that was already considered culturally appropriate.
It incentivises finding the specific animal which was stolen
Sometimes we forget how important the quality of breeding stock is in agrarian environments. Taking somebody's prize sheep or prize ox may very well be worth two regular sheep or oxen. Increasing the cost of the crime when the specific animal is unable to be replaced de-incentivises any crime which may not be 'satisfied' by simply providing two animals of a lesser quality.
What if it can't be paid at all?
This is more of a side-note, but it's worth considering that there is a likelihood that a man who felt the need to steal a sheep or an ox may very well not have multiple oxen/sheep to repay the crime with. It's plausible that incurring a debt of four or five animals in many cases would go beyond the criminal themselves.
Perhaps in practice this would cause a near-relative of theirs to go into debt to satisfy the price, causing the criminal to bear the shame and cost of that to their own family. Or if a family were to refuse to pay, then this same criminal would likely then go into debt/bondservitude to the victim or somebody else in the community to satisfy the price.
Conclusion
It's possible that the higher price is based on the increase supposed 'criminality' of the criminal, and is perhaps to deter people from attempting greater and more significant crimes due to the greater and more significant punishment. There are other angles worth weighing around the quality/value of each specific animal, and what the resultant impact on the community might be - but unfortunately much of this depends on speculation and considering other similar systems in historic societies.
Personally I lean towards thinking this is added restitution to restore peace with the victim and/or community - an extra price to be sure that the victim and the whole community sees the moral debt as 'satisfied', aside from the additional shame which would come with the event. But as we are so far outside of this culture and period, it could easily be any combination of these factors or others.
