I wouldn't necessary go so far as to say he's "popular." (Although, personally, I find him quite cute.) This is a highly subjective question, and there's really no good answer, but I'll give it a go, in a "Why do people like Sealand [if they do]?" manner:
1) It depends on what you mean by Sealand's 'personality.' How would you describe it? "Annoying" is a very broad adjective, without much connection to circumstance.
2) A lot of fanwork involving Sealand/Peter is directly connected to England/Arthur--not surprising by any means, given their history together during World War II and Sealand's subsequent development thereafter. Arthur Kirkland is one of the most referenced (and probably thereby one of the most popular) characters in the entire series, and his relationship with Peter does seem to mirror that of his most famous charge, Alfred.
3) Additionally, Peter's been featured in multiple strips, and has come to have relationships of various degrees with other well-liked characters, including his adopted parents--Berwald/Sweden and Tino/Finland (and by proxy, Hanatamago, Raivis/Latvia, and Iceland.
4) With that much exposure, and with the connections to both Alfred and Arthur, it wouldn't be surprising that some of the fanbase would feel a slight fondness for Peter, regardless of his childish and childlike behavior. He's only twelve, after all. A recent study found that we tend to anthropomorphize even inanimate objects (in this case, like countries), and consequentially interact with them the same way we interact with human beings, even if we consciously understand that they do and will not react in the ways in which we expect. If fans do treat Peter like a 'real person' and thus express feelings of fondness towards him, it's possible that they probably do the same to small, cute but belligerent children in their non-virtual lives.
It's also possible that most of fandom has a tendency to be very forgiving towards figures that are mainly portrayed in a positive light--and to my knowledge, Sealand/Peter hasn't done or said anything in particular that would cause fandom to turn against him. In fact, some might see Peter as one of the more innocent and relatable characters of the series: in addition to his young age, his personality leans toward a hopeful and unrepentant optimism (not unlike Alfred), he has little to no real responsibilities (as Berwald and Tino and occasionally Arthur take care of him), he's carefree and enjoys being around people--but also seems like the type to have mischievous thoughts. (He was known for pirate radio way back when and he IS blood-related to Arthur, after all.) When you take those characteristics into account and match it with the characteristics of the new fandom (mostly pre-teens/teenage, with some university/post-education), one would almost expect a mild affection toward someone like Sealand/Peter.
Lastly, while I did my best to try and answer your question, I think a better discussion would be to analyze why Arthur is such a popular character, when in real life, it'd be difficult to get along with someone so hot-headed and tsundere. (Personally, I adore him to death. I know equally-if-not-more difficult people in my 'real' life, so it's not as hard for me to imagine befriending someone as volatile as Arthur.)