The nominative case "er" maintains the grammar and the logic of the sentence, because "als" is a conjunction, not a preposition. "than him" is typical of colloquial English, which puts pronouns into the accusative case when they follow anything that looks at all like a preposition.
Psssst. The English translation is not grammatically correct tho. It should be "I have fewer coins than he does," or more colloquially, "I have fewer coins than him." (Also correct but old-fashioned sounding: "I have fewer coins than he.")
In sentences like this, the second clause avoids repeating the main verb by just using the auxiliary. Here the first verb "have" is a main verb in the present simple, so the corresponding third-person singular auxiliary should be "does." One only uses "have/has" as the auxiliary when the main verb is in a perfect tense. For example: "I've eaten far more burgers than he has." Or, to keep the main verb "have/has": "I've had more luck than he has."
The problem isn't "having coins." The problem is that if the first clause has "have" in the present simple, the second clause needs "do/does" for the auxiliary. The same rule applies whether you're "having coins" or "eating burgers."
If you reeeeeallly want to talk about having coins and use "has" in the second clause, you can, but the only grammatically correct way is to use a perfect tense, like this: "I've got fewer coins than he has." Like, seriously, check out the lessons on auxiliary verbs in Cambridge Grammar in Use.
To put it another way, if you say the default translation out loud, people will understand your intended meaning fine, even if some might notice that it's not quite right. But if you write it like that on an IELTS Writing Task 1, you'll get some cred for trying to make a complex sentence with a comparative, but you'll get docked for failing to do it successfully because the auxiliary should've been "does." If a sentence like that showed up on the multiple choice sections, "has" and "does" would certainly both be answer choices, but only "does" would be marked correct.
So the German sentence in this exercise can really mean that he has more coins than I do, or that I have the same amount of coins as he does? How do you differentiate then?
Of course they are both correct English, but isn't the question here whether they are correct translations for the exercise?
No, this sentence means that I have fewer coins than he does.
Scorbett's original question was about which of two sentences were valid English. Both examples are valid.
There's then the separate question about which sentence is a correct translation of the German sentence. The default translation is correct.