I guess an oversimplified answer is that he's got a complex about not being his real parent and the reason Jun doesn't have a "normal" mother and father couple. He seems to be an especially insecure person, and feels pretty inferior to his older sister (Jun's biological mother) and people Jun's dad dated before, and even Jun's dad himself. So he hides in a costume and doesn't speak so that Jun doesn't have to find out who he really is yet.
Although, yeah, objectively, it's also weird for his parent to always be in a chicken costume, he worries that Jun will experience discomfort knowing that he's a man and also that he may reject him for it, or for not being his biological parent etc.
He seems to be cis. As far as the pronouns go, I'm actually not sure whether or not they are using gender specific ones, since Japanese also has gender neutral pronouns, but English translators are often forced to choose a gender (and if they're calling him his "mother" and Jun doesn't know his real gender then using female pronouns would make sense). Additionally, and I'm less sure of this, but I gather that some pronouns aren't gender-specific but rather can get used by anyone, but are generally strongly associated with one gender. In not sure of the nuance there, and how strange or eccentric it would be to use them unconventionally, but it's possible that he uses more feminine pronouns himself more as an indication of how he sees himself as a weaker, less forceful and confident person (yeah, gender stereotypes meh). Basically, it's hard to tell how much is them outright using the opposite gender pronouns in order to hide from Jun that he's a man, and how much is just the lack of variation an nuance available in English pronouns.
Anyone know why the mum always wears a chicken costume and doesn't speak? And if the mum is a man why do they call him her?
I don't think he is trans right?