The way that they do it is strange
Why would "maja" not work for this sentence?
Because it's "we" here, and "mają" is "they".
Could this be my mamy chleb? I just wrote mamy chleb and it worked. Or is it just awkward to say?
Yes, it can be "My mamy chleb", but "my" is just totally redundant, because the form "mamy" makes it perfectly clear what the subject of the sentence is.