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I am a little confused if this sentence is grammatical or not:
Assuming that any object can be recognized by detecting their primitive shapes, a new approach is developed.
Would "any" be appropriate here, or should I use "their" or "its"? Or should I use another word entirely?
I'm not sure why the question title focusses on the word "any", but let that pass. The main fault in OP's sentence is the use of singular they for an inanimate object. Per that link, even if some pedants still disapprove, you can use "singular they/their" for a single person...
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...and you can use it for multiple inanimate objects...
Much-loved books can be recognised by their well-thumbed pages.
...but you can't use it for a single inanimate object...
✲I recognised my book by their well-thumbed pages.
So OP's sentence has to be...
Assuming any object can be recognized by detecting its primitive shapes, [something can be done].
Note that I removed the first "that" because it's superfluous (but not actually incorrect). And I didn't transcribe the words after "shapes", because I think that part is badly written anyway. But there's nothing inherently wrong with plural "shapes" (a single object may be composed of two or more primitive shapes).
"Any" can be used with either singular or plural. You can say, for example, "You may choose any one box" or "You may choose any two boxes."
The problem with your example sentence is the combination of the singular noun, "object", with a plural adjective, "their". As the sentence appears to be considering objects one at a time, it probably should read, "Any object can be recognized by detecting its primitive shape ..." It would also be acceptable to write, "Any objects can be recognized by detecting their primitive shapes ..."
Also, "assuming" in this sentence should have an object. If what you mean is that you are assuming that the objects can be recognized, remove the comma: "Assuming that any object can be recognized by detecting its primitive shape, a new approach will be developed." If there is some other assumption here and the ability to recognize objects is the result of that assumption, then you should state it after the word "assuming", like "Assuming that the objects are [whatever], then any object can be recognized ..."
