It is possible that shellac was used on your paneling back in the 30's (or since then, for that matter). Shellac was commonly used back then - but so was lacquer, varnishes, rubbing oils, beeswax, wood polishing (burnishing), and I'm sure others...the telltale sign of shellac though is it's high gloss. Your board doesn't appear to have much, if any, sheen to it - and if there's still a surface coating that remains, I think the lemon oil wouldn't be able to penetrate completely as you've described.
So, there would be no incompatibility with a penetrating oil and a remnant of a shellac (or other) film, but any shellac could keep the oil from penetrating...and these oils are too soft to serve as a surface coating. The best recommendation really would be to test some out on an inconspicuous area and see what happens - you should be able to tell immediately if the oil is going to penetrate - if it just lays on the surface (not what you've experienced with lemon oil), more prep would be needed to then finish with penetrating oils.
Mike mentioned a solvent wash of either mineral spirits or alcohol or both to remove any waxes that may have been applied over the years. Built up waxes could also impede the oil's ability to penetrate...typically waxes will show a build up, in less handled areas of your board, that may be a little soft or gummy. The more wax applications over the years, the higher sheen will be apparent in those areas that aren't handled or cleaned as easily as the flat portion of the boards (especially inside corners).
If you were to consider a french polish with shellac - the finish appears hand-rubbed (although a higher sheen than a penetrating oil), because it is hand rubbed. Unfortunately, it's not as easy of an app as is oils. It requires mixing of components (shellac/alcohol/turpentine) and spreading with a rag folded to about the size of your fist. There are youtube videos to watch that'll help, but it can be a little tricky. The results, on the other hand, are beautiful - stunning - exquisite - amazing. It's actually kind of a lost art. Kinda labor intensive though...
I shoulda mentioned earlier that Danish oil comes in many different "tones"...Natural is probably what you'd be most interested in - it is clear, but has an amber cast (as most oils will), that brings out a depth and richness to natural wood. Regarding the differences between the other oil type finishes available through Watco - If there tung oil is actually a tung oil varnish, I think it'd be more a surface app than a deep penetrating oil...if it's straight tung oil, it would actually penetrate and harden the wood, and have more moisture resistance than Danish oils have, but i'm not sure that'd be of benefit to you. The rubbing varnishes are not something I've used a lot of - they can look great, but they are a surface coating without a great deal of penetration.
I'm not sure if I answered your question or made things more confusing, but I mean well when I'm rambling...