Wasn’t/isn’t there a campaign to get it changed to Sheffield? Which is why it often come up with Sheffield on post things? Maybe it’s because they think the stigma of living in Barnsley is a worse thing than having a phallus named town that is world famous for having that word in its name…. This picture shows what Google thinks…
That’s got nothing to do with residents ‘asking to be part of Sheffield’. As I mentioned before, all Google maps is doing is using the Postcode and address data from Royal Mail. S36 postcodes have the ‘post town’ of Sheffield, hence Penistone, Sheffield. Barnsley is the ‘post town’ for all S70 - S75 Postcodes, but there are places with Barnsley as the post town that aren’t in Barnsley (e.g. South Hiendley which is actually Wakefield but has an S72 Postcode). Royal Mail’s address data and post towns don’t completely match with metropolitan boroughs and their boundaries.