It's very hard to tell a high-solids opaque stain from a low-sheen paint; however, there should be primer under the finish if it's paint, unless it was painted in the last few years with one of the new self-priming products like Duration or Aura.
It's certainly true that stain will soak into the wood more than primer or paint, but the depth of penetration could be difficult to determine, especially in denser wood.
It sounds like Karl knows a lot about the subject, but his first comment can't be taken as an absolute indicator by any means. As I implied above, some high-solids stains these days are almost as thick as paint and, when multiple coats have been applied, can fill depressions in the wood, smoothing the surface much as paint would.
Answer for Identifying solid color stain.
It will be difficult to tell the difference if you are not used to these different types of finishes. Here is a list of differences;
1- Solid stain doesn't cover up or fill the woods texture. A paint does fill most of the finer texture and some larger cracks.
2- Stains have very little or no sheen after full cure.
3- If you scrape the surface with a fingernail a paint will act like a coating or film. A stain will not feel like a film.
4- A stain, solid or semi-transparent, soaks into the wood. A paint sits on top of the wood. Solid stains, especially acrylics, don't soak in as much but still are different than a paint.