In biology there exists a hierarchy of biological classifications into eight major taxonomic ranks along with rules to define new members of the taxonomy and provide the proper naming according to conventions. This system provides a framework that can be used to find and add new information resulting from research.
I have seen specific partial taxonomies within certain mathematical sub-domains but no overarching system like you would see in biology.
Although the AMS Mathematics Subject Classification database can be searched by subject classification (very unstructured), it isn't very useful for finding objects using morphological descriptions.
It has been time consuming for me to determine whether or not certain morphological objects are currently defined, what mathematical domains use them, and when defining a new object, what its name and classification should be.
Can someone please provide me with a source reference for such a classification system for mathematics.
Thanks for your help.
No such classification exists.
Let's look at terminology first. To begin with, I offer as evidence of no rigid terminological system the fact that each of the following are well established terms in mathematics:
the snark;
green fields, red fields, and emerald points;
and my personal favorite, the majestic iterable weasel.
As far as terminological rules go, there are some "local" taxonomical systems - evident in such terms as "hyperhypersimple" - but these systems are (a) mere convention (nobody argues that "computably bounded" is incorrect as an alternate name for "hyperimmune free"; at best, people prefer one name to another for aesthetic reasons) and also don't cut across subdisciplines at all (e.g. "simple" in computability theory has nothing to do with "simple" in group theory). Really, they're the exception rather than the rule (and as a fun exercise, try to find as many different technical meanings for the word "tame" as you can - I suspect you can get into the triple digits without much difficulty). At the same time, there are words which mean the same thing across mathematics, but there's no system for coming up with these, and e.g. the fact that "set" means something technical while "collection" does not is reflective of nothing but a historical word choice.
Now if we pass to trying to taxonomize mathematical subfields, there are lots of organizational schemes which exist (e.g. the AMS classification you mention). However, each of these are purely descriptive: they describe the existing state of affairs, rather than prescribe how future mathematics should be organized. Indeed the most interesting mathematics is that which cuts across established subfield boundaries!
Note that this is true even in biology: the taxonomies you describe apply not to the subfields directly, but rather to the objects of study. Taxonomies of mathematical objects indeed exist (e.g. the classification of finite simple groups). However, there is no serious taxonomy of all mathematical objects, nor should there be: for any taxonomy you can write down, I can come up with some mathematical object which doesn't fit it! This reflects a fundamental difference between math and biology: biology studies an existing (and hence limited) collection of objects, and one of its many goals is to establish a classification of these. On the other hand, math doesn't really do that (and indeed the question "What is a reasonable object of study for mathematics?" has been hotly debated throughout history). The idea of a complete taxonomy of mathematics isn't (I would argue) inherently interesting, and when a new mathematical object is discovered, we're really more interested in what it is, and describing it and its properties accurately and clearly, than we are in fitting it into a preconceived picture of what sort of mathematical objects there are. (Which is not to say we ignore existing organizational ideas, just that we don't view them as particularly fundamental unless they are useful/interesting in a given context. E.g. when a group theorist defines some new class of groups, they usually don't bother to say whether this is a first-order definition, or infinitary first-order, or second-order, or . . .)
To pick just one example of this "dispersed" attitude, note that you will often here that "ZFC is the foundation of mathematics," yet most mathematicians cannot name all the axioms of ZFC. Nor should they necessarily be able to - it's often not relevant for what they do. You may view this as horrid disorganization; I view it instead as the right organizational approach to what is a very different field from biology!