My mother wishes my child be like me.

The way this would be taken by me would be that the mother causes the child to be like the parent throught the transformative process of wishing—“Wishes” being a transitive verb. (But even then, to be a complete sentence it should be “My mother wishes my child to be like me.”)

I believe it should be “My mother wishes for my child to be like me.” To wish for something would not be taken as transitive.

Without being a native English speaker, I'm pretty sure karl B is right about the subjunctive mode. It's not colloquial for sure, but I did have a professor who had studied at Cambridge, picked up a strutty attitude, and insisted on speaking in subjunctive mode. "Since we assumed x be a value greater than..."

I think most people would say "My mother wishes my child would be like me" (or ...could...etc.); however, "My mother wishes my child be like me", with "be" replacing "is" to reflect a wish for something that is not presently so, is correctly using the subjunctive, as you mentioned. There are quite a few other ways the same sentiment could be expressed, but that doesn't make your example wrong.

It sounds extremely archaic to me.

The "wish" verb does require the subjunctive . . . but the form of the "be" needed is "were." And isn't there another issue here?? "Me" should be "I" with the implication of "like I am."

My mother wishes my child were like I (am).

And there are my two cents for today!

Isn't the sentence "My mother wishes my child were like I" more of a conditional mode rather than a subjunctive mode?
Conditional mode generally implies the impossibility of attaining the goal stated:
"If I were a rich man" (As the song says).
Whereas the subjunctive implies a will and a possibility to attain a goal. In this sense it is closer to the imperative of a verb.
Also another uncertainty here is the tense of the verb were vs. be.
"my mother wishes my child were like I" would seem to refer more to a child who already exists, whereas "my mother wishes my child be like I" could imply a child yet to be born, as well as the possibility that only if I do the right things the child will actually be like I.

"like" used as a conjunction (that is, used to introduce a clause as in "like I am") is seen by some (like Strunk & White) as illiterate. It isn't, tho, it's standard English. But, "like I" sounds weird to me, I prefer either "like me" or "like I am".

The "were" and "be" here are both considered the subjunctive mood. I think "be" is more archaic. And in speech we use "was" instead of "were" (My mother wishes my child was like me). I've heard that for some speakers, "were" instead of "was" implies a lower likelihood.

"Be" isn't more archaic. "Be" is the present subjunctive and "were" is the past subjunctive. Normally, a wish for something contrary to fact, as in the example being discussed, would use the past subjunctive: “My mother wishes my child were like me.” The present subjunctive is used with "that" clauses for commands, intentions, necessity, not wishes. For example: "My mother insists that my child be like me." Both are examples of the subjunctive, but one states merely a desire the other a demand.

Porsche, yes. But "be" after "wish" or "if" (where we'd use "were/was" nowadays) sounds archaic to me. For instance: "If music be the food of love, play on."