1. 'any of a number of songbirds' is what it should read, Jack-- did you copy it correctly? 'Which' introduces a modifying clause for 'songbirds'. 'A number of' is a premodifier of 'songbirds'; it means 'several'.
2. and 4. are incorrect grammar.
5. and 6. It should be spelled 'anisotropic'-- 'in physics, the quality of exhibiting properties with different values when measured along axes in different directions. Anisotropy is most easily observed in single crystals of solid elements or compounds, in which atoms, ions, or molecules are arranged in regular lattices.' (5) is the correct mode of expression: 'measurement' is an uncountable concept in these definitions.
7. and 8. refer to the meaning(s) of words in the dictionary. In (7) the phrase is referring to several meanings of a word in the dictionary; in (8) it refers to only one meaning of a word in the dictionary.
9. This I presume came from a dictionary or other brief reference; these, like newspaper headlines, often do not contain insignificant words like articles and the verb 'be'. You can put in the article (which would be 'the', not 'a') if you like, and it does not change the meaning, it just makes the sentence more appropriate for an essay or more formal statment of the definition of 'tautology'.