A layman's question: Given a sentence whose meaning is unique, is there any standard algorithm to uniquely determine/describe its grammatical structure such that it can be classified in a standard way (i.e. like, if I don't err, what exists in fields like organic chemistry etc.)? If yes, please give some good references. Thanks in advance.
There are multiple competing procedures.
Linguistics is still quite far from proving true even fractions of the grammar of any specific natural language. Even the most basic phenomena are interpreted differently in various paradigms. For the most obvious example: one important fraction within theoretical syntax, and at least historically the most important one, is Generative Grammar in a Chomskian tradition. For example, GG relies on strictly binary branching trees and assumes every noun phrase carries "case". This approach is all but marginal in computational linguistics, where algorithms are created to parse sentences; instead, computational linguistics favors dependency grammar and constructionist approaches. So at the very least, syntactical theory and computational practice are so far anything but unified, or in agreement about a standard.
What does exist are some subfield standards which cover, if at all, only aspects of what you're asking for. For example, WALS employs a standard coding scheme, and constituency tests are a near-standardized syntactical toolkit for establishing what things should be bracketed together. There are also powerful computational tools for classifying sentences according to one specific grammar, and some parsers count as quasi-gold standards to compare other algorithms to. Yet, a GG syntactican would reject many of these analyses as being all but irrelevant for "real" syntax.
However, none of these are uncontroversial even within these subfields, and none of them come close to providing a unified standard for exhaustively describing the syntactic structure of any sentence.
Also see this question.
I do not know whether there is any sentence in the world that has a unique meaning. I am not sure I can identify all the things that contribute to the meaning of a sentence, since it depends on context, on the way it is stated or written, on the way people interpret various signs that go with it. Even single words seldom have a unique meaning. Meaning may also depend on the mood of whoever produces or receives the sentence.
Regarding algorithms to determine grammatical structure, they are dependent on how you define the grammatical structure of a language, and that is very far from being standardized (or to put it differently, there are competing ones). It is that definition that matters, not whatever algorithm you may produce to recognize the structure.
It is actually unlikely that any single description will explain all structures of a language. It is probably as unlikely that two people would agree on what belong or does not belong to a language. Furthermore, any language you consider is constantly evolving in time, and may vary geographically (and dead languages changed over time, so that remaining documents do not necessarily represent the same time or space variant).
In other word, it is a lot fuzzier than chemistry, and even organic chemistry, because the human brain is a very big and complicated machine. This fuzzyness is somewhat underscored by the fact that a good part of the work on natural language processing is based on statistical techniques.
For the time being, the best we can hope for is analyzing a variety of linguistic phenomena, and try to condense them into theories that will account for most, expecially the more frequent.
Grammatical structures may be more complex than trees, depending on what you wish to include. But even ignoring that, I would not really see the point of considering together sentences with the same tree structure as a special class. What matters in the theoretical support that assign such structure to sentences.
I would guess that most textbooks on natural language processing will have a chapter about describing the structure of sentences, though it may change some, depending on the book. This does not mean in any way that the knowledge is random, or totally disorganized. There are many bridges between the various approaches, and various ways to preserve coherence. As I said, the human brain is big and complicated.
If you are interested in formalization (as you seem to be), there is a considerable amount of formal results in computational linguistics. But this does not mean it is able to accurately describe all aspects of natural language.