The use of personal arms by the Swedish royals is actually regulated by law. Basically, as far as I understand,
what the law says is that the Greater Coat of Arms of Sweden coincides with the King's personal coat of arms and that, in addition, the King can give "other members of the Royal House" permission to use the greater national coat of arms as their personal arms with such changes and additions that the head of state determines. My understanding is that the State Herald advises the King on the design of the personal coats of arms of members of the Royal House.
There was some confusion regarding Prince Julian because, when his siblings were stripped of the HRH (before he was born), the Court's press release seemed to suggest that they (and by extension any future child of Prince Carl Philip such as Julian) were no longer considered members of the Royal House. If so, how can Julian have a royal coat of arms?
The answer seems to be that, following the interpretation of the Constitutional Committee of the Swedish Parliament, any reference to the "Royal House" in Swedish law, including the Act of Succession and the Instrument of Government (and, I assume, by extension the law on the Coat of Arms of Sweden) is to be interpreted as a reference to all persons in the line of succession to the throne, regardless of their titles or styles, which the Committee accepted is a matter to be decided by the King on his own discretion.
Therefore, Prince Julian is still a "member of the Royal House" in the strict legal sense and, accordingly, entitled to a personal coat of arms that is a differenced version of the Greater Coat of Arms of Sweden, even though, for internal purposes, including titles and styles, the Royal Court now uses a narrower definition of "Royal House" that does not include Prince Carl Philip's or Princess Madeleine's children.
It is all quite messy to me to be honest. To make things worse, it looks like "the Royal House" also includes Queen Silvia, Prince Daniel and Princess Sofia, as they have also been given (registered) royal coats of arms, even though, following the Constitutional Committee's interpretation, they are not in the line of succession. Maybe we could say that the Royal House includes then people in the line of succession or the King
plus their respective consorts, but that definition fails in the case of Chris O'Neill, who is not a member of the Royal House and does not have a royal coat of arms.
And what about Princess Birgitta? She is also considered a member of the Royal House (by the Court at least) and bears personal arms, but is not (and never has been) in the line of succession either, nor is she a consort of someone who is in the line of succession. Princess Birgitta's arrms were granted, however,
before the current law came into force, so maybe the King at the time had a greater freedom to grant personal coats of arms. Furthermore, the style HRH at the time was borne by all paternal line descendants of King Karl XIV Johan as long as their marriages were dynastic, which is Birgitta's case.