It isn't all the same. Plenty is, absolutely, but plenty more may be processed in the same place, but to different standards, as others have noted. Waitrose weren't caught up in the horsemeat scandal, despite buying from the same places, because their meat was separately processed, from separate supply chains, and the machines cleaned between use for them and for other supermarkets - and they sell fruit and veg that is better stored and so it is riper than some places, but not as quick to rot. Some things are clearly identical, sure, but enough isn't that it's worth it to us.
We buy things like dishwasher tablets, Heinz ketchup (DS can't eat any other sort) etc from Lidl, but fresh food from Waitrose or Ocado. The quality is noticeably different IMO, and we do try it every few months, just to see. Processed food, or cleaning basics, and Lidl is fine. Fresh, and we find the gap in quality not worth the saving. We also buy bulk things from Costco, where quality is also brilliant for fresh fruit and veg, and they also store and keep really well. Their tinfoil is great - but we recently learned that they chop down virgin arboreal forests to make their loo rolls, and it's an environmentally catastrophic option, so we've turned to Who Gives A Crap for those.
We also get milk delivered in glass bottles now, because the single biggest contributor to our plastics bin was from milk. They collect the empties, sterilise and reuse. That costs more but we can swallow the difference, which I appreciate isn't everyone's position.
I think it's about balancing what's worth it to you and what isn't. As well as what you can afford, obviously. It's hard because I know a lot of our choices aren't as environmentally friendly as I would prefer, but we can't afford to be wholly virtuous so we have to just try where we can.
Waitrose is also a worker's co-op. They treat their staff well. Lots of other places very much do not. So there's that, too.