Plato's Idealism seems to imply an objectivity and an anti-pluralism, instead rationalizing forms into main groups yet German Idealism seems to do just the opposite. I was wondering why the use of the same term to describe them both and what the context is.
Well, first of all, Platonic Idealism is often called Platonic Realism. This is because Plato believed that pure ideas exist outside of mind, and that they are in fact only true reality, while world we perceive represents only temporary and fleeting shadows of these eternal ideas. For example, every tree we could see is just a shadow of real idealistic tree. Plato also believed that true reality could only be "touched" trough thought.
Although appealing, Plato's theory is still very much a fairy tale without proof. Over the years, we are not getting closer to his World of Ideas, not only those discovered but those still unknown to us. At the same time, negative parts of his work (knowledge gained empirically could be just shadow or a dream) is still relevant, as is skepticism.
German Idealism (starting from Kant) acknowledged deficiencies of empiricism, and tried to find firm ground for knowledge in mind itself. Similarity between German and Platonic Idealism lie mostly in universality of certain mental categories, which were relatively modest for Kant, and culminates in Hegel's Absolute Mind. Absolute Mind is very similar to Plato's World of Ideas. Chief difference would be that Hegel envisaged Absolute Mind as developing itself (as each individual mind would be a part of it) , while Plato held that his ideas existed outside time and were always perfect .
There is no essence of idealism; there is no common and distinctive nature which all 'idealist' theories share. (The same is true of 'realism'.)
There is merit in the following characterisation despite its age:
idealism historically contains four main propositions: (1) Plato's (value is objective - its meaning and origin lie beyond the human knower); (2) Berkeley's (reality is mental - there is no non- mental being); (3) Hegel's (reality is organic - wholes have properties which their parts do not have); and (4) Lotze's (reality is personal - only persons or selves are real). Any system is idealistic which affirms one or more of these four propositions. (Edgar Sheffield Brightman, 'The Definition of Idealism', The Journal of Philosophy, Vol. 30, No. 16 (Aug. 3, 1933), pp. 429-435: 432.)
I might hesitate over the precise wording in some cases but the strength of this characterisation of idealism is that it displays the requisite diversity. So far as I know, nobody has held all four positions : they are independent, not mutually consistent (Plato rejects Lotze) and this rules out their being mutually entailing.
You need to note, of course, that there is a certain asymmetry in your question. 'Plato' refers to a particular thinker; 'German idealism' extends widely beyond any single thinker. To Hegel we must add Fichte, Schelling, and others. Many commentators would include Kant in 'German idealism' but despite his 'transcendental idealism', which has a fairly technical sense, Kant's Critical philosophy is too complex and synthesises too many elements to labelled in any way. The only case for putting Kant among the 'German idealists' is that he fits, with some affinities, into the late-18th/ early-19th century period generally known as that of 'German idealism'. A weak case, in my view.
I'd add that 'idealism' is not to be identified with and does not entail 'solipsism', the theory that only I exist. This is sometimes referred to as 'subjective idealism'. If an idealist is a solipsist it is by virtue of extra assumptions to the propositions listed above.
My general advice about philosophical labels is to ignore them - peel them off and look at what individual philosophers say. I wholly endorse Mauro's remark : 'It is hard to understand the works of individual philosophers if we restrict the history of philosophy to the genealogy of "isms".' Another general point is that once you use a label, you look for commonality and it may simply not be there. This is one of many lessons we should learn from the later Wittgenstein.