Why was the king of Judah referred to as "the king of the House-of-David"?

Uncertain. But some scholars simply interpret this a synonymous usage for a geographical destination, in this case for Judah.

There are quite a few difficulties to read from that stele. It is rather short, fragmented, the fragments not at all certainly joined at the precise position the earliest editors thought them to be; although they might quite likely have chosen the 'correct' position.

A very short introduction to some aspects of that debate are:

Since 1993–1994, when the first fragment was discovered and published, the Tel Dan stele has been the object of great interest and debate among epigraphers and biblical scholars. Its significance for the biblical version of Israel's past lies particularly in lines 8 and 9, which mention a "king of Israel" and possibly a "house of David". The latter reading is accepted by a majority of scholars but not all.

Dissenting scholars note that word dividers are employed elsewhere throughout the inscription and one would expect to find one between byt and dwd in bytdwd too if the intended reading was "House of David". They contend that reading dwd as "David" is complicated since the word can also mean "uncle" (dōd) (a word with a rather wider meaning in ancient times than it has today), "beloved", or "kettle" (dūd). Lemche and Athas suggests that bytdwd could be a place-name and Athas that it refers to Jerusalem (so that the author might be claiming to have killed the son of the king of Jerusalem, rather than the son of the king from the "house of David").[34] R.G. Lehmann and M. Reichel proposes interpreting the phrase as a reference to the name or epithet of a deity.

According to Anson Rainey the presence or absence of word dividers is normally inconsequential for interpretation.[36] Word dividers as well as compound words are used elsewhere in the inscription and generally in West Semitic languages, so it is possible that the phrase was treated as a compound word combining a personal name with a relational noun. Mykytiuk argues that readings other than "House of David" are unlikely. Yosef Garfinkel has been vocally critical of alternate translations, characterizing them as "suggestions that now seem ridiculous: The Hebrew bytdwd should be read not as the House of David, but as a place named betdwd, in parallel to the well-known place-name Ashdod. Other minimalist suggestions included House of Uncle, House of Kettle and House of Beloved."

Francesca Stavrakopoulou states that even if the inscription refers to a "House of David" it testifies neither to the historicity of David nor to the existence of a 9th-century BCE Judahite kingdom. Garfinkel argues that, combined with archaeological evidence unearthed at Khirbet Qeiyafa, the inscription's reference to a "king of the house of David" constitutes primary evidence that David was a historical figure and the founder of a centralized Iron Age II dynasty.

— Wikipedia: Tel Dan stele

The text is not vocalised, open to substantial interpretation and quite some debate for individual choices, with some of the debate quite heated due to the attached 'consequences' when interpreting this. Missing or present matres lectiones, seen as unusual word boundaries and or line breaks, most seem open to substantial variations:

The inscription raises a series of important issues. The divergences are mirrored in the different reconstructions and translations of the text. I have come across 13 different editions of the text referring to Fragment A, and no less than 20 editions of the joint fragments A and B, taken as a joint or as individually separate fragments. My bibliography has over 170 entries on books and articles in some way or another dealing with the Tel Dan inscription – only 10 years after its discovery!

That means that especially the most famous and sought after word on the stele, bytdwd ('ביתדוד' allegedly 'House of David') presents also quite a room for debating the various interpretations:

G. Athas (p. 248) finds the diphthongs ay, written with y in qdmy (line 5) and aw, written with w in bytdwd (line 9). […]

G. Athas is not able to see any matres lectiones in the inscription (pp. 248–249). This can be questioned against byt in bytdwd, where the yod (٦) is easily seen as an plene written ê (with mater lectionis), compared to the otherwise usual form ht. As Athas (and e.g. Biran and Naveh) reconstructs line 6 B, he sees a possible masc. plural ending in [šb^cn "seventy" (pl of (šb^c), but defective written, with no mater lectionis. If it is correct to reconstruct a nota accusativi (יyt)/ as is possible in lines 7 and 8 and even more certain in the transition between lines 9 and 10, we presuppose a mater lectionis.

If יby is to be reconstructed as יby[l] (line 4, A), there is a medial long Î (Lipihski 1994, pp. 90 and 97), but this is an uncertain reading.

The important word bytdwd and its usage here has thus sometimes found a quite simple solution:

Some scholars will read bytdwd as a geographic designation, which is right if read as synonymous with Judah.

— Hallvard Hagelia: "Philological Issues in the Tel Dan Inscription", in: Lutz Edzard and Jan RetsO (eds): "Current Issues in the Analysis of Semitic Grammar and Lexicon I. Oslo-Göteborg Cooperation 3rd-5th June 2004", Deutsche Morgenländische Gesellschaft Harrassowitz Verlag: Wiesbaden, 2005.