Why is it "Rhine", but "Rhenish"?

Being a native German speaker, I just came across the word "Rhenish" (as a translation of German "rheinisch", belonging to the Rhine).

I am a bit confused about this, and am at a loss for the proper "language words" to describe it... I have long since ceased to think about either German or English in grammatical terms, going by "feel", and it's just not clicking here.

The German transformation from Rhein to rheinisch follows the "usual" pattern for such transformations (e.g. Logik -> logisch).

But the English transformation from Rhine to Rhenish "feels" awkward, and the result looks like a rather mangled attempt to spell the German rheinisch. The dropped / missing "i" looks especially weird. Going with my "gut feeling" I would have expected something like rhineish...?

Sorry for not being able to put my finger on it, but... can somebody explain what is "going on" here with Rhine -> Rhenish?

Sometimes, there are phonological rules that tell you what the sound change should be under a modifications.

But here it just seems to be a historical/cultural choice, not uncommon in English, to choose a alternate, classical derivative for that slot (the adjective version/demonym of a place name):

Rhenish: "of or belonging to the Rhine" (especially of wine), late 14c., from Anglo-French reneis (13c.), from Medieval Latin Rhenensis, from Rhenus (see Rhine).

This is similar to Naples->Neapolitan, Venice->Venetian, Norway->Norwegian.

So there's no native English sound change to explain the short 'e' in 'rhenish'. It was just an academic choice of a different word altogether.

Wiktionary gives Rhinish as an alternative form for Rhenish. There are also Rhenian, Rhenic and Rhenane. According to Wiktionary the etymology is:

From Rhine +‎ -ish (with the first element taking a Latinate form; see Rhenus).

So just in general I guess many words aren't intuitively inflected. The adjective for Wales is Welsh. The adjective describing the geographical area of Flanders is Flemish, and people from there are Flemings. A person from Liverpool is a Liverpudlian. The adjective for Cambridge is not"Cambridgian", but Cantabrigian. God only knows from and through what languages these words have been passed down to us.

If you look at a list of demonyms/ethnonyms you'll see many examples like this.

Actually, the word Rhenish, which ultimately comes from the Latin name of the river Rhenus, shows an older vowel sound than the early New High German Rhein. Both Old High and Middle High German called the river Rîn; it was only in early New High German that the diphthong appeared. And there's also the German surname Rhenisch, which I assume goes back to a monophthong Rhine river. The h that crept into the Latin and NHG words is basically a decoration from the Greek.

Rhenish is primarily used in the UK for any German white wine, which Americans call "Rhine wine." American tastes are now sophisticated enough that those who know a little about wine will distinguish it from one that comes from the Mosel/Moselle.