Quite right... the same way that store branded products come from the same places as the name brands. For example, here in NorCal, Albertson's-brand milk is from the Berkeley Farms dairy. However, there are variations in this... Berkeley Farms processes the milk somewhat differently (or perhaps uses somewhat different-quality milk) for their name brand than they do for the store brand. They do taste somewhat different, although Albertson's milk is just fine by me.

The same is likely true of gas. The "overflow" refinement may not be exactly the same as the branded refinement, not considering the additives, etc... they may not have the same filtration processes to remove water and other impurities beyond the minimum requirements, etc. So, just because it comes from the same refinery doesn't mean it's been refined in exactly the same way. Small differences but over time they can be noticed, as I mentioned.

(Note that this doesn't mean I am endorsing name-brand over no-name. I buy store brand products all the time. I'm simply commenting that it's not necessarily the same manufacturing/refining procedure even if it is from the same place. Whether the difference is significant and/or noticeable and/or you care about it is a totally different issue. )