Saying something is akin to something else IS a metaphor. Analogy, simile, whatever. It's a kind of metaphor. You've again made the mistake of thinking an analogy is an equivalence. Then you're claiming it's a "false" equivalence and is therefore trivializing. I think we can handle analogies.
Also, as far as I can tell without a paywall (!?), neither definition you linked to mentions beliefs or feelings so I'm not sure how you're getting that. I don't disagree that stereotypes can cause prejudice, I just don't follow the distinction you're trying to draw between beliefs and feelings or how it makes a difference here.
The problem OP was trying to highlight is people making unjustified assumptions about people based on their pre-conceived notions (which are also indefensibly unmoored from reality).
Racial prejudice. Which adds a boatload of social and historical context to "prejudice" alone. Not to mention that no one talks about racial prejudice without (at least) an awareness of the discrimination and oppression that such prejudice often gives rise to.
> This isn't a teachable moment to explore the nuances of racism, it's simply a metaphor.
"Tell[ing] me what I must like [on the basis of star sign]...is akin to racism" (OP's words) isn't a metaphor. It's simply stating that one thing is analogous to another. I realize this all seems totally pedantic, but making a false equivalence between an astrological stereotype and racism trivializes the experiences of people who are on the receiving end of racial discrimination.
> I think stereotypes are a kind of prejudice, again, using plain definitions.
If we're simply looking up dictionary definitions, I'm happy to point you to the dictionary-defined distinction between stereotype (beliefs about characteristics of groups)[1] and prejudice (negative feelings towards groups)[2]. That being said, you did just repeat my point that stereotypes often form the basis of prejudices :).
[1] https://www.oed.com/search/dictionary/?scope=Entries&q=preju...
[2] https://www.oed.com/search/dictionary/?scope=Entries&q=stere...