Hi Zachary
I really can't be sure if it's having an adverse affect, but I wouldn't be surprised if it was.
Having looked at just 3 of the product pages, there is a problem with content being repeated, but I think it is being compounded by there being no other content on the page either to make it look unique.
Both are the hallmarks to a potential Panda penalty, which could affect the pages performance themselves and/or the whole domain. So, if you're seeing subpar performance (and even if you're performing well it's worth reading on) I would look at the following solution.
For every product that you do not intend to restock or reuse, I would either add a tag, add a 301 redirect or simply remove the page and serve a 404. If we're talking tens of thousands, then having that many redirects might bloat out your .htaccess file (making it larger and longer to load/process) and having an instant drop of 20k URLs and 404 errors might look a bit odd to Google as well. However, adding 20k tags is a bit of a nightmare as well.
You might want to try a combination of all 3 - a few 404 errors is nothing to worry about - but the logic is that you will be removing a number of pages that have this duplicate content on it, thus improving the quality of the domain. For your remaining 'live' pages, I'd highly recommend taking the time to add 200+ words of unique content about the product in order to avoid this happening again.
An alternative solution would be to block the bots from accessing the /shop/ subfolder in your robots.txt file - and then setting up the shop and the currently active product listings on a different subdomain. You'd lose the ability to use the /shop/ folder, but it would be quicker than manually adding tags or 301 redirects.
That's the method I would use if I wanted to address the issue. However, this **may not be necessary **if your site is not performing badly. You can check this to a degree with the Panguin Tool - this overlays your organic traffic in analytics with Google updates - if a drop in traffic coincides with a Panda penalty, you may be under the affect of one - in which case you should take action ASAP.
Hope this helps.