It could be misconstrued that Netflix is endorsing it and because there is an alternative way to describe what Netflix does (i.e. Rental Service), they may be asked to change the wording. If using "Netflix" is the only way to describe the product without actually implying endorsement, than they will probably be safe. Another perspective is that you can generally use other trademarked names as long as you don't make profit from their actual names. So I can make a netflixsucks.com site and would be protected in doing so as long as I don't make a profit. Other than that, as long as they are truthful and don't confuse customers they can probably find a way to finagle using "Netflix for Baby Clothes" somewhere in their promo material.

My opinion is that this is a clear case of confusion. That a layman could readily confuse the origin of the goods with the mentioned company. You usually have to acknowledge the RTM/TM and make a clear statement that your goods do not originate with the TM owner and are not warranted by them.

If you can't afford to answer a lawsuit then I'd avoid it! You don't need to be in the wrong to get sued.

Comparison advertising is usually specifically allowed and the form of comparison advertising makes clear that your goods do not originate with the company you're comparing with. Just don't lie especially if you chose a large company to compare with, they'll have you for every last cent.

While it's an interesting hypothetical, I don't actually see the word "Netflix" anywhere on the site. A Google search suggests it's a snappy description they've used when describing the site to a journalist, but I don't see that they've written it anywhere themselves.

If they had done, well...I think you'd have to be very very careful. You wouldn't want to create the slightest hint of confusion that your service is endorsed by Netflix.

It's mentioned in the video and the Facebook "like" button.