Henry IV is not unique within Shakespeare for having two plot lines running side by side. What is unique is that at this point, it’s very hard to see how they will ever have anything to do with one another. It’s very easy to see in Twelfth Night that when Sebastian returns, there will be confusion between him and Viola, or that Richard returning to Wales will obviously have to encounter Bolingbroke at some point. With the two plot lines in Henry IV, we get very little insight in the first two acts as to how they will be related in a practical way. However, in a thematic way, the ties between the two story lines are there.
For instance, early on King Henry admits that he is jealous of Henry Percy for having a child as impressive as Hotspur.
Yea, there thou mak’st me sad, and mak’st me sin
In envy that my lord Northumberland
Should be the father to so blest a son – …
Whilst I …
See riot and dishonour stain the brow
Of my young Harry. (1.1.77-85)
Hotspur has nobly fought the Scots and imprisoned many.. Meanwhile, Harry is hanging out with thieves and commoners. What ties these two characters together is both this line and, with the threat of rebellion from Hotspur becoming apparent later, that both Harry and Hotspur are potential heirs to the throne. It’s also worth pointing out that Harry and Poins have essentially overthrown Falstaff and the others, or at least sneakily rebelled against them (though only for laughs, evidently) while Hotspur is planning to rebel against the King. The two seem to have completely contrasting characters yet are similar enough in their predicaments to see this contrast as being something deliberate. Perhaps Shakespeare is going to say something about leadership with these two characters, or the difference in behavior of the privileged and someone who must fight for power. I’m not sure if there is anything conclusive to say about this relationship in just the first two acts, but at this point I’ll be reading on with the idea that the contrast between these two characters is worth noting.