*you mean kingship?

Well,

the important thing about Late Medieval and Early Modern culture in this regard is naturality in externals. They were aware of the possibility of a misleading exterior, but they believed such a thing was unnatural. Properly speaking, the exterior should match the interior, according to God's design and natural law things being as they are made. This anxiety is represented in Hamlet: "I have heard of your paintings too, well enough; God hath given you one face, and you make yourselves another." This quote invokes the suspicion at the time that makeup might be sinful as a kind of vanity, perhaps even idolatrous as a misdirection from God's design. You also have the theory of correspondence— a twisted body means a warped mind— which suspected most deformed persons of being evil, and also often involved suspicions on the mother as well.

Clothing was associated with rank as well. It was considered very important than nobility were able to look as grand as they were in status. They even introduced Statutes of Apparel in Henry VIII's time onwards; which hoped to regulate this and have clear limits on the extravagance of middling aspirants, to mixed effectiveness. Materials, colours of dye, the extent and complexity of patterning, fur, feathers, jewellery and adornments; these would have all been understood as having various degrees of fineness and nobility.

In addition, the Medieval and Early Modern concept of privacy was not solitude primarily, which was quite uncommon, but more exclusion. The privilege of being able to choose who you were with in certain rooms, and the exclusive status of those rooms to inner circles. I don't know if the garb you are referring to is appropriate to his rank in general given I haven't seen the movie yet. But wearing casual clothes in the presence of inferiors could be potentially subversive. It would be a mark of substantial trust and intimacy. So not impossible, especially given Shakespeare's characterisation which they are using, but it depends on where he is as to how likely. Oddly, it would be more plausible to be seen in casual clothes by inferiors by letting them into the bedchamber, which was exclusive but far from solitary, than to be seen in informal public situations casually. In public royals would be very well dressed, and they don't really do informal by our standards; given the monarch's public appearances are all essentially stage-plays. Even in the bedchamber scenario, the intimacy of seeing them that way is a performative gesture, and the royal would still be expected to behave themselves according to rank and decorum. Monarchs could of course push the boundaries there, or even get away with breaking them, but it matters. Again though, this is potentially justifiable with Shakespeare's characterisation.

As to whether the real Henry V would have, well, Shakespeare's characterisation is exaggerated and based on rumours as much as solid information. So probably not anything too bold. The real Hal made conscious efforts to protect his reputation by affecting model behaviour. And he had in fact been involved in politics and war from early on. So again, not ruling it out, but anything too "base" or indecorous would have been out of the question, even in his younger days.