I was reading about the idea of a specific colour as an abstract object as defined by Plato, and how in 'Parmenides' he struggled with the fact that the type cannot be single and exist in multiple places without being 'split apart'. I am interested in why some believe that such abstract objects need a physical location when their instances only need to, why did Plato and further on Russell discuss the idea of the location of abstracts as if they had physical sense?
Abstract objects have puzzled philosophers for millennia, and there is not a current consensus on how to deal with them.
One "troubling" aspect of abstractions is that they appear not to be material, or physical. They constitute no mass nor energy, and may not even have any actual instances in history. This is "troubling" for the simpler forms of materialism and physicalism, as then abstractions would not exist -- yet they certainly seem to. Most philosophers who are physicalists today have adapted physicalism to accept that not everything in the universe IS physical -- IE physicalism is not a monism. At least this is the case according to Daniel Stoljar in his relatively recent book "Physicalism". This relaxation of physicalism to no longer be a monism is also apparent in Kim (Physicalism or Something Near Enough), Papineau (The Rise of Physicalism), and Melnyk (A Physicalist Manifesto). None of these authors focus on abstract objects, so they don't provide answers to your questions -- but they illustrate how the dominant ontology in philosophy today is still struggling with this issue, and does not have a straightforward answer to it.
The two dominate approaches to abstractions are abstract realism, and nominalism. A lot of physicalists used to lean toward nominalism, as that appears to give a path to maintain physicalism as a monism. But that is not really clearly the case. And that is particularly true now that most physicalists have fully adopted emergence as a valid phenomenon in our world. Where are "nominal" operations taking place, other than in a logic space? Even if "we" create temporary logic space to do a useful nominal exercise -- isn't' that just another word for an only temporary emergence of an abstract space and objects? And what is the "we" who is doing this "nominally"? Emergent physicalists today have generally accepted the reality of consciousness and selfhood, and hold them to be emergent non-physical phenomenon, just with a physical DEPENDENCE to them. Neither selfhood nor abstractions need to be only "nominal" if physicalists abandon monism for "dependence", and this tactic has drastically reduced the need for physicalists to adopt nominalism. Nominalism can therefore be seen as a rationalization tool that physicalism resorted to when it tried to maintain itself as a monism, but has now lost most of its popularity.
The clearest ontological description of abstract realism I have found is in Popper's adaption of Three Worlds from Frege. https://tannerlectures.utah.edu/_resources/documents/a-to-z/p/popper80.pdf Most philosophers today have accepted that abstract objects are real, and exist. This is the case for math realism, and for moral realism. The general claim that all abstractions exist -- is less clear. Most physicalists today have implicitly accepted a world 1 plus world 3 dualism as the basic nature of their universe, with consciousness as an emergent phenomenon from world 1. This is a very similar ontology to Popper's but emergent non-monal physicalism is more suspicious of the independent reality of consciousness than Popper is. Popper's three world emergent triplism itself also grants less independence to consciousness than does Cartesian spiritual dualism. But the differences between these views all focus on tweaks to how they address consciousness, NOT abstract objects. Most philosophy today is fully compatible with abstract object realism, despite the discomfort or queasiness that many philosophers feel about adopting a fully populated world 3 ontology.