What might be the negative legal ramifications of adjudicating a case in US courts that deals with the customs and traditions of immigrants?

While the law would provide for penalties if such discrimination can be proven, the process of proving it would require that the plaintiff identify themselves as part of a group that has no definition or status in current law.

First of all - not at all. In california the law bans discrimination actual or perceived. See for example this informational poster from the CA government and Moore v. The Regents of the Univ of Cal (2016):

Therefore, a pretextual termination of a perceived-as-disabled employee's employment in lieu of providing reasonable accommodation or engaging in the interactive process does not provide an employer a reprieve from claims for failure to accommodate and failure to engage in the interactive process.

To be illegal, the discrimination doesn't need to be based on factually correct attributes. It being motivated by these attributes, whether actual or perceived, is enough. The claimant doesn't need to admit to these attributes, and they may not even be true at all. What the claimant needs to prove (as Deborah Moore did in the case cited above) is that the discrimination is rooted in the discriminator's belief they possess the attributes.

As to a group that has no definition or status in current law - anti-discriminatory laws define the groups, and this group would be defined as well in the proposed legislation.

Indeed, it is quite likely that anyone immigrating from a society with such a system would desire to be free of it, and may have done so with the intent of shedding all remnants of such an oppressive system.

This may be true, but the dominant caste members also immigrate to the US, and bring their prejudices with them. Which is exactly the motivation for the legislation proposed in Seattle, and several other places with large immigrant populations of South Asian origin.

While noble in intent, it felt somehow wrong to even acknowledge such divisions, and got me wondering if there might be any downsides to legally recognizing an oppressive system that does not exist in US law, and in fact directly contradicts it.

Prohibiting discriminatory behavior doesn't mean it is being condoned (in fact, it is the exact opposite), and whether the law recognizes it or not doesn't affect its existence.

canada

Law can recognize new categories of discrimination.

The claimant in a discrimination case need not establish or concede that they in fact possess the characteristic against which they allege discrimination occured.

The claimant could allege something like

D has discriminated against me on the basis of their belief that I am of X caste.

I myself reject the classification of people into castes. This makes D's discrimination on this basis all the more harmful.

See School District No. 44 (North Vancouver) v. Jubran, 2005 BCCA 201:

Azmi Jubran, who does not identify himself as homosexual, was repeatedly subject[ed] to insults and harassment of a homophobic nature during the five years he spent in high school.

...

It is common ground that a person who is perceived to have the characteristics of a person who falls within one of the enumerated grounds in s. 8 may be the object of discrimination although he does not actually have those characteristics.

...

The homophobic taunts directed against Mr. Jubran attributed to him the negative perceptions, myths and stereotypes attributed to homosexuals. His harassers created an environment in which his dignity and full participation in school life was denied because the negative characteristics his harassers associated with homosexuality were attributed to him.

...

The words of s. 8, as they apply to discrimination on the ground of sexual orientation, do not require that a person complaining of discrimination identify himself as a homosexual.

See e.g:

It is discrimination to evict a person ... because the landlord believes they are First Nations ancestry

See the Manitoba Human Rights Code, a purpose of which is:

to restrict unreasonable discrimination against individuals, including discrimination based on stereotypes or generalizations about groups with whom they are or are thought to be associated