At one level, I get your point about taking time off and wanting some compensation. The process is burdensome and can be time consuming. However, this just isn't the way things are done anywhere I know of. Yes, you are taking time off (unpaid apparently) to visit with them; at the same time, they are taking time out of their normal business to meet with you. It's a shared risk, since it may prove you are not suitable them, or they are not a good place for you to work. Thus, you should both share in that risk: You go without pay for that day (or use a vacation day or make up the time), they don't get any of their normal work done. Furthermore, you'll only do this with them once, while they'll probably have to do it a few times.
As pointed out in Jim's answer, if the company pays you for the time off, then they'd have to pay every candidate they interview. From their perspective, that would set a bad precedent (as people might line up for interviews just to get paid), and could become very costly if they need to interview several candidates for a single job. So, it's pretty much unimaginable that a company would actually pay you to come in for an interview.
In response to your edit saying you are out of vacation time:
This isn't the problem of the company interviewing you. In fact, if you tell them this, their response might be that you seem like someone who can't manage their time and that they don't think you're a good candidate for the job anymore. And no, it wouldn't be any different if it was an hourly job. Would it be possible for you to make up the time somehow - either by working on what would normally be a non-work day, or by working extra over a few days?
Update: As pointed out in the comments, being out of vacation at the end of the year (when this question was originally posted) may not be a sign of bad choices by the OP, but a wise use because of the current employer's policy. While not every employer will make new vacation available right on January 1, some do. If your employer is one of those, you can ask for the interview to be delayed until after that date, possibly using the holiday season as a reason for the delay, assuming the holidays celebrated in Europe and the western hemisphere (Christmas, Hannakuh, Kwanzaa, and New Year's) are observed where you are.
Final note: Overall, it sounds like you're not really interested in this company. If that's accurate, save yourself and them this waste of time and cancel the interview.