The short version, although probably not enough for someone who would ask the question:
だ is a contraction of である, which functions as a pseudo-copula. It is a combination of the particle で and the verb ある. There needs to be a noun preceding it, which will then function as the で-marked "argument" of ある.
だ is not in the same category as sentence-ending particles like ね and よ. It has clear grammatical function - although it is re-grouping things in a way that obscures the natural pattern. The polite form of だ is です, itself a contraction of であります. (However, です has evolved a separate usage, which also allows it to be used at the end of a sentence after i-adjectives and certain conjugated forms of verbs.)
In the example 私は彼女だね, ね is a colloquial sentence ending, adding the nuance of ", right?", as noted. The core of the sentence, however, is 私は彼女である. Contracting to だ (or です formally) at the end of a sentence is more or less mandatory, but the analysis will be easier to follow using the un-contracted form である - so all the examples below will do that.
Here, we functionally have a verb ある - "to exist" - with a topic 私 ("I") marked with は, and a... mode of existence? marked with で. This sentence is about the speaker, and it asserts that something exists in a "her" way. In other words, that something "is" her. Although the で is normally contracted with the verb, it is really a particle that marks the preceding noun's role.
Many observations at this point:
There isn't an expressed grammatical subject here (marked with が). Japanese is a "pro-drop" language, where these things can formally be omitted (i.e., it is grammatically correct to do so) and the listener is expected to infer them from context. English struggles with both of those: a grammatical subject almost always needs to be stated explicitly (even if that subject is "it").
In particular, 私 isn't the thing that 彼女である - that would mean "I am her". By using a topic marker instead, we allow for something else to be the thing that is identified as "her". Why do we have a sentence that is about 私, when 私 isn't the thing that ある? Er... good question. (If you have a few years to research, start by looking up the classic example 「[象]{ぞう}は[鼻]{はな}が[長]{なが}い」.)
The English verb "to be" is very strange. It can mean "to exist"; but it can also function as a copula, which we understand as either relating an adjective to a subject predicatively ("the cat is cute"), or predicatively equating (in multiple different ways, actually) a subject to a direct object: "This animal is a cat" (categorizing), "That is my cat" (identifying), etc.
Japanese doesn't have anything like that. The closest thing in Japanese equivalents to an "adjective" can freely be used predicatively already: 可愛い猫 ("cute cat") -> 猫が可愛い ("the cat is cute"). For equating things predicatively, Japanese uses the verb ある, which is pretty close to "to exist" (there is also いる, but we won't get into that now and we don't use it in this context), with the particle で to indicate the role.
The example cannot be properly understood as "omitting が好き", because that would be changing the で-marked part from 彼女 to 好き, and similarly changing the role of 彼女. Although, contextually, 私は彼女が好きである could mean the same thing, it conveys that meaning in a different way.
Note here that 好き is grammatically a noun, not an adjective - although it is one with very restricted uses. Practically, we only find it either before だ/です, or used as a "na-adjective" (but see the note at the end). In English grammatical terms, 好き is basically a gerund formed from the verb form 好く - although directly using that verb is so unidiomatic as to be almost theoretical-only.
Anyway, the point is that 私は彼女が好きである means that "she" exists in a "liking" way, and this sentence is about the speaker. The natural inference is that the speaker is the one doing the liking.
What about the original?
The sentence is about the speaker, and there is something identified as being "her". We also know from context that the speaker is claiming to like her. What is the implied subject? It might, therefore, be "a liked person": 私は好きな人が彼女である. In this rendering of the subject, 好きな人, we get to see the use of the combination 好きな as a "na-adjective".
For bonus marks: using な to turn a noun into an adjective like this is really the same thing as using だ=である to do it. (There are complex historical reasons why な is used instead of だ in this context; basically, Japanese used to conjugate verbs differently in these two contexts, and dropped the distinction in most cases but kept it for these idiomatic, contracted forms.) However, sometimes we have to use の instead, which is functionally different and probably better understood as "just a way to connect nouns".
