First some background information, as you are noticing a pattern.
Generally speaking a "Notice of DMCA removal from Google Search" is submitted by a 3rd company that claims to be specialized in finding infringing content and submitting requests to remove it. Such notices and removals of links from Google search are not the initiative of Google or based on Google's own bots or spiders.
However, the companies that are specializing in collecting URL's to infringing content often simply have bots or spiders running over the internet, querying search engines with broad generally infringing terms, and note down every URL that has closely matching keywords (read content/text), and is not yet white listed by their system.
Thereafter, they automatically generate reports in Google's required format, and submit a removal request.
Google thereafter has a (human or automatic - not sure) process to verify the claims, and if in doubt, removes the links to the content without warning.
This in turn also affects your SEO rankings, as a large percentage of removed URL's vs. total URL's of your website, could mean you are a copyright infringing website.
How to counter notice
Google has a fairly simple Counter Notice system and it works well. Generally speaking when you submit a counter notice, which can be done by following the link from the e-mail where Google notifies you of a DMCA removal, Google follows up within 48 hours. In this form of Google, you fill out why you believe/claim the removal was in error. After submission of your counter notice, Google confirms that they received your request. This request is then sent to the company that asked for the removal of your URL. They always reply, as they are obliged to, if they don't reply within time (14 days), Google re-instates your URL within generally 10 days after that (this is from experience, not from what's claimed on the internet). If the company does reply, they could likely ask you for further information, ask for your legal contact, vice versa. In the end if your URL was requested to be removed in error, the company could white list your URL, this is up to the sole discretion of the company and the strength of your claim and your ability to prove that you will not likely have infringing content on your site.
Your site has user generated content. This is something often red flagged by such companies and hard to white list. Do you have proven processes mechanisms that control user generated content before it appears on the web? This is perhaps one of the most important considerations for companies to allow white listing you. Do you have a good record just like YouTube has? Another important factor, before starting to compare yourself with YouTube.
I hope this answer helped, and good luck reinstating your URL's.