In an analysis of a sentence, a noun has one grammatical function. This means that when a linguist describes a sentence then each word is described as an subject, object, complement etc.
But this does not mean that there are no ambiguous sentences. In "Time flies like an arrow/ Fruit flies like a banana" the word "flies is first a verb, and second a noun.
Moreover different grammarians may parse a sentence differently. The "rules" of English are (like any other language) not perfectly understood and different people will sometimes disagree on what the correct analysis should be. An in-depth reading of (for example) The Cambridge Grammar of the English Language and A Comprehensive Grammar of the English Language will find differences in opinion and analysis. Many of these differences will be minor.
In you particular example, the Noun phrase "spider climbing up the stairs" is the complement of "like" and in this noun phrase, "spider" is the noun that heads the phrase, and "climbing up the stairs" is a participle phrase that modifies "spider".