One name, that too flagship product name.
1. Building one brand is hard enough, having two splits focus. Most likely your company name won't be much mentioned @ social media and your marketing team will not bother to keep that website amazing as that does nothing to Popsoles sales.
2. Press will be confused when speaking with you. Are you company X or Popsoles? Sure you can explain but real estate is limited and one is better than two. Look at nest for example. Even providing a website address will be a question- should we mentionpopsoles.com
. Eg: Thalmic Labs/GetMyo started with two. But they had to consolidate and make one website as two names were confusing in press etc. Even for users, they want to talk to the product brand, not company behind.
3. Hiring will be confusing and you will struggle as most of your focus will be on building Popsoles and that will have the press coverage and the social media clout. But @ LinkedIn you will be hiring as companyX and people won't find a facebook page of CompanyX with thousands/hundreds of thousands of likes.
4. It feels weird initially to have the product name also be company name (esp for the starting team as they dont have the outsiders perspective), but its normal. Its classic manufacturer mindset. "PopSoles is a footwear technology company that makes..." sounds very normal to me. Users don't even think twice. You are Popsoles. Get a crackin logo though. Don't cut corners in that investment.
5. Have a company page and use that to do the company positioning. Talk about all that the company stands for, vision, mission, purpose, future roadmap direction etc.
7. The argument that if the product brand has bad reputation it will not affect the company is not valid for startups. If shit hits the fan, get a new name and logo and start again. If PopSoles takes a hit, having a separate company doesn't matter as users won't touch PopSoles, so the company brand is just an fantasy idea (and a website somewhere in the internet universe). You live and die with PopSoles. And in future if you are big and have multi-product lines, in that case cross the bridge when you get there. Most startups don't reach that level for years, over planning is also a thing. Sure people might think you are one trick pony (ie only one product company, but that kinda thing investors care about, not consumers. If PopSoles rock, consumers don't care if you have a product line up or just a single hot product). Investor care about traction. Get traction with PopSoles, share product roadmap with potential investors via company page.
8. You will save trivial discussions with your design team. With your marketing team. With your HR team. With your mother in laws neighbour.