First of all, it sounds like there might be some confusion regarding the claim.
As I read it, the claim is not that steel is inherently better than plastic. It's not an apples-to-apples comparison. The claim is that one reusable steel bottle is better than continually buying many disposable plastic bottles. Certainly, one steel bottle has a much bigger impact than one plastic bottle, but if you buy a nice $10 stainless steel water bottle, hopefully it'll get used over and over again.
As you might suspect, this depends on many factors. You can probably find many analyses online, but all of them will have to make some assumption about the rate at which bottles are recycled. That may not be relevant to your decision. You probably know how likely you are to recycle plastic bottles, which might be much more than the average person. For that average person, giving them a shiny possession (the steel bottle) might help discourage them from throwing away a seemingly cheap plastic bottle.
One reasonable write-up of this issue was one I found on the NY Times site. Their conclusion was:
What it comes down to is this: if your stainless steel bottle takes
the place of 50 plastic bottles, the climate is better off, and if it
gets used 500 times, it beats plastic in all the environment-impact
categories studied in a life cycle assessment.
Can you reuse a steel bottle 50 times (the climate change balance point)? Or do you lose bottles/lids too easily (or do like me and drive off with them on top of your car)? And, again, do you religiously recycle plastic bottles, or only do so if a recycle bin is convenient? Also, I'm going to plug another question I asked here, as I think it's relevant, if you recycle via public bins.
Also, try to remember that in general, with sustainability, you want to try to reduce consumption first, if you can't do that, reuse resources second, and only if that's not practical, recycle last. Recycling is an imperfect process, especially with plastic bottles.
Finally, it's worth noting that although not strictly a sustainability issue, many believe stainless steel bottles to be better for your health, as a result of fewer harmful substances leaching into your water, compared to plastic bottles.