I would do analytics to identify which of these sites performs better in search and then put a big effort into making a massive improvement in content. This effort would be to make both sites absolutely different from one another and the site that is getting this effort absolutely superior to just about any site that is out there.
If these sites perform differently in different niches, I would divide the effort being placed into them niche-by-niche.
I would start with the most profitable niche and focus on the product pages on that site first, then move down on the basis of profit priority. I would make the best product pages possible, detailing all of the important features and giving tips for use that might stimulate sales. You are not just beefing up for the search engines, you are beefing up to benefit the customers.
I would also do an assessment of which products generate the most customer service questions. For those I would produce informative content that answers those questions in great detail. This would not be skimpy blog posts. It would be substantive articles with photos and data that clearly answer the questions. The goal is to produce a library of this content. Then customer service people can simply refer people with questions to those pages. Some people will find those pages on their own and that should reduce the customer service effort. This should produce great content that will be shared by customers and save you employee time.
I would also look into products and pages that are unproductive. These are dead weight on your site and not worth the time needed to improve them. This is a good time to prune and weed.
1. Is it possible to get the same product pages on page 1 of the SERPs for both sites?
Yes, but you will benefit more by getting one page to the top of page one rather than getting two on the bottom of page one. My goal would be getting one page to rank as high as possible.
2. Should we be canonicalizing (is that a word?) products across the sites? This would get tricky - both sites have roughly the same domain authority, but in different niches. Certain products and keywords naturally rank better on 1 site or the other depending on the niche.
The intended use for canonical is to attribute identical content. My focus on this would be to make massive improvement of the content of one site.
If you use canonical attibuting product ABC to just one site then you will have only one site in the SERPs.