There are two things worth noting here. First, in present-day English, "show" is usually a transitive verb, but Shakespeare uses it here as an intransitive verb. Hence "shows like" here means "looks like".
See E. A. Abbott: A Shakespearian Grammar, 1884, § 293 ("Transitive verbs are rarely used intransitively"), which gives several examples of this, including Bushy's lines from Act 2, scene 2.
The other aspect is that either Shakespeare or the typesetters of the Folio often used a third person plural in -s. See Abbott §333: "Third person plural in -s".
Another example cited by Abbott can be found in Act 3, scene 2 of Hamlet (italics from Abbott):
The great man down, you mark his favourites flies,
The poor advanced makes friends of enemies.
Interestingly, the Oxford Shakespeare, the Cambridge Shakespeare, the Penguin Shakespeare and the RSC Shakespeare don't amend "shows" and "is" to "show" and "are", respectively. What is actually happening is that "which" here means "each of which", and the singular forms "shows" and "is" depend on "each" rather than "which". (The antecedent of "which" is "shadows".)
Bushy is saying that each real sorrow brings twenty illusory ones with it, each of which is not a real one. (The contrast between appearance and reality also appears elsewhere in Shakespeare's works. See for example, '“Seems,” madam? Nay, it is. I know not “seems.”' in Hamlet, Act 1, scene 2.)
The impression of twenty shadows is caused by what Bushy describes next:
For sorrow's eye, glazed with blinding tears,
Divides one thing entire to many objects