I wrote my undergraduate thesis on Irish folk music and its role in politics, national identity, and violence through the lens of The Troubles. I did not use Danny Boy in my research for a couple of reasons, but I think this frame of reference is useful for the sake of your question.

I see your question in two parts:

1. Why is this song considered to be Irish when does not seem to be about Ireland?
2. Why do people even like it?

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I am actually going to answer these two components in two posts because I am, it seems, very wordy.

1. Why is this song considered to be Irish when it does not seem to be about Ireland?

An Irish folk song does not have to be written by an Irish person in order to be an Irish folk song. "Dirty Old Town," recorded by The Dubliners and The Pogues, is widely considered to be an Irish song, though it was written by Ewan MacColl (an Englishman born to Scottish parents) about an English town. (Slightly alarmist article)

Despite the salacious headline of the article, Irish songs being less that 100% Irish - as if such a thing could be measured - is not a scandal. "Leaving of Liverpool" is about a journey from England to California, yet listening to The High Kings' 2011 recording of "Leaving of Liverpool" from Live in Ireland," an entire audience of Irish concertgoers sings along (2:45). According to my observations in my undergraduate research and in my conversations with Irish folk musicians and enthusiasts, the metric of an Irish folk song has less to do with the origin and more to do with the listener. A few illustrative points:

• American folk music has its origins in foreign and diverse musical traditions, yet we, the consumers and performers, broadly call it "American." From PBS:

At the beginning of the 20th Century, the term "folk music" was used by scholars to describe music made by whites of European ancestry, often in the relatively isolated rural South. As the century progressed, the definition of folk music expanded to include the song styles - particularly the blues - of Southern blacks as well ... Increasingly, music made by other groups of Americans such as Native Americans, Mexican-Americans, and Cajuns came under the umbrella of "folk music."

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• Music from Canadian artists like Drake and Justin Bieber pass as "American" in many circles.
• The Pogues' recording of "The Band Played Waltzing Matilda" is considered to be an Irish song because they are an Irish band (formed by Hiberno-Brits in England) though it is written by a Scottish man about an Australian service member in Turkey in WWI. It can be found on such sites as irishsongs.com and featured on historyireland.com. These are not academic sources, rather the reflection of the opinion of a populace, which is what really designates a song as a local phenomenon or not.

Irish music is incredibly syncretic, with instruments like violins, concertinas, and bouzoukis being considered traditional, or "trad," instruments. The boundaries of what make a song "Irish" are incredibly vague.

Addressing the Scottish influence in your question directly, music has passed freely across the borders of Ireland and Scotland for many, many years, and enough qualities are shared by both traditions that it can be difficult - and useless - to try to separate the two when they overlap or share a blurred boundary. The following is a block quote of my own work, which I do not know how to cite in this sub, but I will include relevant in-text citations below:

Luke Kelly sings I’m A Rover in a truly excessive Scottish accent he sometimes favors. It’s unclear, however, if the song is actually Scottish in origin. I’m a Rover’s chorus, “I’m a rover, seldom sober” is mirrored in the line “Cause I'm drunk today and I'm seldom sober/A handsome rover from town to town” from Carrickfergus, which is about a Northern Irish town.^(1) I’m a Rover shares the lyrics “I will be guided without a stumble” and “Who’s at my bedroom window/ Disturbing me at my long night’s rest” with The Night Visiting Song, another Luke Kelly tune whose origins weave through England and Scotland.^(2) Despite the uncertain origin of I’m a Rover and The Night Visiting Song, both songs sport versions calling the lady love “Molly Bawn,” which is both a popular and demonstrably Irish ballad and a common Irish reference.^(3) Additionally, when Scottish singer Davie Stewart picked up the tune in Ireland, neither he nor Scottish folklorist Hamish Henderson mentioned Scotland at all.^(4)

I have not read about Frederic Weatherly having a relationship with Scotland, but it is alleged that he had not ever visited Ireland, and many confuse features of the two countries.

So, why is this song considered to be Irish when it doesn't seem Irish?

Because Irish folk music, like many other musical traditions, is syncretic, and many songs, especially ones that originate nearby, are absorbed into the popular canon.

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1: The Dubliners, “I’m a Rover,” track 7 on Seven Drunken Nights, Starline, 1971, LP; The Dubliners, Carrickfergus, track 6 on Now, RAM Records Limited, 1975, LP

2: The Dubliners, “The Night Visiting Song,” track 12 on Alive and Well, RAM Records Limited, 1972, LP; Davie Stewart, Davie Stewart, Liner notes by Hamish Henderson, Topic Records, 1978, Vinyl

3: Jennifer J O’Connor, “The Irish Origins and Variations of the Ballad ‘Molly Bawn,’” Canadian Journal for Traditional Music 14 (1986): 10–18, 26; Davie Stewart, “I’m Often Drunk and I’m Seldom Sober,” Davie Stewart, Topic Records, Vinyl

4: Davie Stewart, Davie Stewart, Liner notes by Hamish Henderson, Topic Records, Vinyl; John Reilly, “Adieu to All True Lovers,” track 1 on The Bonny Green Tree, Topic Records, 1978, Vinyl; Davie Stewart, “I’m Often Drunk and I’m Seldom Sober,” Davie Stewart, Topic Records, Vinyl