If it is wood, I've seen people use water on floors - called by some "water-popping" because it tends to open the grain to accept stain better. It doesn't actually look like wood to me.

Are you sure you got down to raw wood. That looks as if the stain is sitting on top...as if it is sitting on a finish that didn't come off.

Another concern is: certain woods do NOT like taking stain. Yet another concern: if this was every polished with an oil/wax furniture polish, that wax/oil may have penetrated deep into the wood (it leached into the wood through the finish) and you may have to sand much deeper to get past this contaminated wood.

So many things could be happening here.

Look at the same location but on the back - does the top look like it is plywood? If so, then that strip is edgebanding on the end of the plywood. Some shops use vinyl edgebanding that is colored - will not take a stain.

A different direction - if it is wood or vinyl, a gel stain sits on top more than a stain that soaks in (like staining a fiberglass door). You might try that type of stain, just make sure to clear coat it with a compatible clear.