i don't think you have to. if it's not visible from above, it's not a big deal.

if the handrail bothers you, you could always paint it, or stain it darker.

As Beth points out, there is no need to match the floors from one level to another. The important thing is to match the stairs (up or down) the the MAIN living space. What I refer to as the public spaces. These are spaces your guests are going to see. Usually basements have very few guests going through it....which means you can have whatever floor you want.

And if the colour discrepancy between the handrail and the basement floors, you can paint or stain a colour that works well with BOTH floors....just as Beth mentions.

How about two wood floors that abut each other? I have medium toned hardwood oak recently installed in an entry hall that runs into the LR which is now carpeted. I want to replace the carpet with wood/vinyl laminate but not sure how to or not to match the two floors.

I have laminatout of a dog run all daye that looks like slate on my main floor and LVP that looks like wood in my walkout I think they both work for different reasons ans when you get down to it function is first. We put the VP in the walkout for the dogs, they run in and out of a dig run all day in all kinds of weather.