This is one of the features of the dominant accent of large parts of northern England (as well as occurring in some London accents). Many of the people who talk like this are very smart, very highly educated and write beautifully. It would be a shame and a mistake to think we can accurately determine much about the qualities of a person by isolating a few features of accent or pronunciation. English comes in many flavours, and by and large the speakers of all those flavours deserve to be treated with respect and an open mind.

It's a dialect, especially in Yorkshire. It's not 'bad English, people only really speak like this and wouldn't write it (apart from maybe something like a text message conversation with friends or social media).

They are saying “my” but some of their accents make it sound like they are saying “me” instead.

With some it's just bad English. I grew up in a place where "me" was common instead of "my"; it wasn't pronunciation, just really bad English. Sad but true, native speaker does not equate with competence. :(

You’re correct — it’s just a different pronunciation. /mi:/ is a more conservative pronunciation of the word “my”. Nowadays, the standard pronunciation of "my" is /mai/.