Short answer----No.

And, what is oak stained walnut???? Never heard of that. Did you mean walnut stained oak?

Reasons. Poplar is often partly green naturally. That reason, plus being a close grained smooth surface wood is why many tradespeople call poplar the Painters Wood.

Staining wood is never the same. Wood from the same tree can take stain differently. And getting different woods to look similar even with the same stain takes months if not years of
experience.

Plus, you will need to remove the risers to get the paint off.

Plus, poplar stained does not look like oak stained.

Best bang for the look is to remove the risers and replace them with whatever wood the treads are ----and good luck getting those to match.----because not only do different boards of the same species stain differently, what ever finish is used can also change the color.

You could try doing an oak faux finish on the risers. I would research if you could try multiple stain colors rather than paint and do some testing on the same type of wood. There are tools to make a grain pattern. I haven't tried this but am going to experiment on my treads so I can use a runner instead of carpet, with the edges simulating wood.

Here is a link that might be useful: Grain Tool

You could just, as you suggest, paint them. There's no rule that says risers have to be white. A darker color complementary to your treads that will better hide the inevitable toe scuffs on the risers would be a much better choice.

What color or finish is the other trim around the stairs and room

Want a real good job? Apply oak veneer to the poplar and stain and finish as you do the treads. If done by an expert, you will not be able to tell that they are veneer on poplar. You will have to remove the paint to get veneer on properly. That can be done without removing the risers, but you need the professional tools to do that.

Other options are to replace the risers with solid oak or face them with 1/8" door skin. 1/8"door skin is similar to veneer, but is thicker by about 3/32".