I think this may be a good way to remember this: ähnlich is 'similar', and when used in a sentence like the one above, it is 'similar to'. (Actually i think the word 'zu' or 'mit' or something is missing here, although i'm not sure which word, if any, is missing.) So the other part automatically goes to dative.
There are several possibilities to parse this. The most simple one being to view 'jemandem ähnlich sein' as a phrasal verb ('to be similar to someone').
The term 'separable verb' refers to something else, e.g. you have the verb 'anzeigen' which becomes 'ich zeige jemanden an': an affix is removed from the verb and placed somewhere else in the sentence.
It's nothing that's specific to phrasal verbs. Certain verbs require certain cases. Sometimes the meaning changes according to which case you use. 'jemandem ähnlich sein' requires the dative case (as indicated by the 'jemandem'). Equivalently, you could use 'jemandem ähneln' which also requires dative and avoids the construction with 'sein'. Your alternate way of parsing it should also be valid. From a practical point of view, I think it's a good idea to view such constructions as phrasal verbs and learn the whole expression ('jemandem/einer Sache ähnlich sein') as one big chunk of vocabulary. Learn it once, never get it wrong again.
It seems I can't reply under your post, wataya, so I'll just stick it here.
So, phrasal verbs can require a different case than the main verb of the construct, even if there is no preposition involved.
Or we could parse it differently and have ähnlich ("similar") as the core of the predicate and "einer Maus" as a predicative complement. And the combination of predicate nucleus and the predicative complement's case would dictate a meaning, which could vary with case; and not all cases may be valid with all possible nuclei. In portuguese we have something similar that we call "regência nominal", but instead of cases we have prepositions.
I think might have overthought this...
I think the approach "einer Maus ähnlich"="similar to a mouse" mentioned in another thread actually reflects the reality of the language best. It is the adjective that "expects" to be developed by a dative construction. "Sein" just happens to be the verb in the sentence. "Werden" could be another. (I am not confident enough to suggest other verbs; "scheinen" perhaps?)
Flowers for Algernon: June 23—I’ve given up using the typewriter completely. My co-ordination is bad. I feel that I’m moving slower and slower. Had a terrible shock today. I picked up a copy of an article I used in my research, Krueger’s Uber psychische Ganzheit, to see if it would help me understand what I had done. First I thought there was something wrong with my eyes. Then I realized I could no longer read German. I tested myself in other languages. All gone. This is nightmare
The article "einer" is in dative form because "maus" is a die word, and whenever you put an adjective and a dative form together, whether it's an article or a personal pronoun like "mir", the adjective comes last. This is one of the things to keep in mind about the structural differences between English and German, so like for example saying "important to me" would be "mir wichtig" in German, not "wichtig mir" like it would be with English sentence structure. However, it would also be correct to translate it as "wichtig für mich", in which the case the adjective wouldn't come last since it's not using dative form.
There's an extra article. If you meant "alike" then that would not be correct either, because like I've pointed out to someone else here, "alike" requires at least two subjects, so you could say "you and the mouse are alike", but saying "you are alike a mouse" isn't proper English.