The scholastic theologians occasionally address the question of whether Jesus had a guardian angel. The near-consensus is that he didn't, although they rarely spend much time on it.
First, it's worth considering why they took there to be guardian angels, and what they are. Matthew 18:10, in which Jesus says that the angels of little ones are always before the face of the Father, is a big contributor, of course, and Bonaventure mentions also references in Tobit (5:16) and Acts (12:15); St. Jerome explicitly claims that each human being receives a guardian angel at birth, and notes this as an indication of the dignity of the human soul.
Aquinas argues that one would expect this as a part of the normal working of providence (ST 1.113.1):
According to the plan of Divine Providence, we find that in all things the movable and variable are moved and regulated by the immovable and invariable; as all corporeal things by immovable spiritual substances, and the inferior bodies by the superior which are invariable in substance. We ourselves also are regulated as regards conclusions, about which we may have various opinions, by the principles which we hold in an invariable manner. It is moreover manifest that as regards things to be done human knowledge and affection can vary and fail from good in many ways; and so it was necessary that angels should be deputed for the guardianship of men, in order to regulate them and move them to good.
On Aquinas's view, everything in the perishable realm has angelic guardians. The unusual feature of human beings is that we get individual guardians, whereas other things receive guardians for the whole species. This is due to the peculiar nature of the human rational soul and its individual incorruptibility. Each human being is in a sense guarded as if he or she were a whole species. Angels come in different orders; individual guardianship is given to Angels, the lowest order, but more general and universal guardianship (e.g., of nations) to Archangels and Principalities.
Thomas is not very explicit, but his argument seems strongly to suggest that each individual human being has a different guardian angel. However, he notes at least one definite exception: children in the womb share their mother's guardian angel [ST 1.113.5 ad 3]; I'm not sure what St. Bonaventure's view of this would be. Bonaventure has the view that through time humans being share guardian angels, but that at a given time each human being might have only one guardian angel. The general agreement is that guardian angels are only guardian angels in this life, but that's not the whole story; St. Thomas holds that each saint "will have an angel to reign with him" in heaven (ST 1.113.4).
A guardian angel is appointed for custodia (protection) or sometimes exercitium (training); St. Thomas holds that the primary aim is enlightenment and instruction (ST 1.113.5 ad 2), although in some cases, especially with children, guardian angels take a more active hand in defending against demons and physical and spiritual harm. Based on Daniel (10:13) and St. Gregory the Great's comments on it, St. Thomas holds that our guardian angels can be in a sort of indirect opposition to each other. As angels, they are in agreement that God's will may be fulfilled, but they do not necessarily have the same perspective on what that would be, because angels only know those things about God's providence that they think to consider (ST 1.113.8). St. Bonaventure has a view along the same lines, but as usual likes to put things into trinities. The primary reason we have guardian angels, he holds, is to defend against the Devil, who seduces reason by cunning, attracts the will by flattery, and oppresses the virtues by violence against the body. Thus guardian angels are there to instruct the reason, to exhort the will, and to defend the virtues. We need custody because we are outmatched without it, and this would have been true even in the state of innocence before the Fall.
Given all this, one's first impulse might be to think that Jesus also would have had a guardian angel, but as noted before, theologians are near-unanimous in thinking he did not. The fundamental problem is that of custodia. As St. Thomas puts it,
Christ as man was guided immediately by the Word of God: wherefore He needed not be guarded by an angel. Again as regards His soul, He was a comprehensor, although in regard to His passible body, He was a wayfarer. In this latter respect it was right that He should have not a guardian angel as superior to Him, but a ministering angel as inferior to Him. Whence it is written (Matthew 4:11) that "angels came and ministered to Him."
This is about almost anything anyone says on the subject, but St. Bonaventure goes a bit further in his Sentences commentary (2 d11 a1 q3) in developing the idea. He divides his argument into two components: first, Christ did not need custodia, and second, it was not in agreement with His status.
First, as to need, Christ was the Divine Word, and thus Christ's human intellect was illumined by the Word, so He had no need for angelic instruction as part of His defense and protection. Christ's human will was confirmed with the fullness of grace, so He had no need for His will to be exhorted. And Christ's body is united with the divine nature, and therefore He required no angelic defense of it against violence that could threaten His virtues.
Second, as to appropriateness, custodia implies a kind of praesidentia. This is literally true -- both custodia and praesidentia can mean 'guardianship' in Latin -- but obviously St. Bonaventure is making a tautology. It's tempting just to transliterate praesidentia as 'presidence' or 'presidency', but this is perhaps misleading. I think Bonaventure is almost certainly reading it etymologically at 'priority in sitting'; to have custodia over someone, you have to have a higher seat than someone -- to guard someone in a matter of strength, you have to have eminence over them in matters of strength; to guard someone in a matter of understanding, as a teacher might, you have to have a higher understanding; to guard someone in a matter of virtue, you have to have a higher place with respect to that virtue. And, of course, the problem with Christ being in angelic custody shows up immediately: There is no seat higher than the Divine Throne, and there is no angel who has preeminence in any matter over the King of Angels.
Thus Bonaventure says, no angel has custody of Christ, but a multitude of angels minister to Christ. This is how he (and everyone else) deals with apparent places in Scripture suggesting that angels did something guardian-like with Jesus, like the Angel that supported Him in the garden of Gethsemane. This was angelic ministry, not angelic custody. In a sense, we can say that guardian angels help those in their custodia because they are acting on an authority higher than ours (namely, the authority of God, who appointed them), whereas angels aided Christ on His own authority and at His will.
One argument that requires a bit more work to deal with is the argument from dignity. St. Jerome had explicitly linked angelic custody to the dignity of the human soul, and obviously Jesus does not have a soul with less dignity than we do. However, Bonaventure argues that we can distinguish between things that indicate dignity itself and things that indicate dignity in the midst of indignity or weakness, and guardian angels are the latter. That is to say, that was have guardian angels shows that despite our weaknesses, limitations, and defects we have a relatively greater dignity than other creatures who share our weaknesses and limitations. The saints in heaven, however, have no guardian angels because they have a dignity even greater than that, and this, of course, is also the case with Christ.
If you were to ask what most Catholics meant by 'guardian angel', both today and for the past several centuries, I suspect you would not find them making a strong distinction between custody and ministry, so if one spoke more loosely, one could perhaps say that Christ has a guardian angel -- indeed, probably many -- in that they give him ministerial service (as an armor-bearer fortifies the soldier, as Bonaventure puts it). But in the strict sense, guardian angels exercise custodia, which would not apply in the case of Christ.
There are some broader ramifications of this. In the Protestant Reformation, many Protestants became skeptical of the notion of guardian angels, despite references to 'their angels' or 'his angel' in Scripture. But what they argue against is specifically custodia; the early Calvinists (like Calvin in his commentary on Luke), for instance, hold that all the angels minister to all the elect as part of their service to Christ, so essentially the argument is that just as Christ has angelic ministry and not angelic custody, so also we. While the Counter-Reformation did involve many concessions to Protestant concerns, it also established points at which Catholics held the line despite criticism, and guardian angels were, practically speaking, one of them. The early Protestant opposition to the notion of guardian angels, however, has arguably been slowly crumbling for about half a century now, and I suspect it's for the same reason that most Catholics would not hesitate to say that Christ had a guardian angel, namely, that the rank-and-file don't really make a strong distinction between custody and ministry, nor, for that matter, between individual and collective guardianship. The common line of thought tends toward, 'Were there angels? Yes. Were they helping out? Yes. Then guardian angels.'