i was hoping to gift this to a friend, but I don't want to give something that has a substance that could potentially be toxic! Why is it greasy when it's just wood, and is the oil toxic?
There are a few likely contenders for the oil used on this but given the construction my money would be on it being a non-drying oil, specifically mineral oil or liquid paraffin (very similar to the base for baby oil, just a little thicker).
If it's finished in a known wood-finishing oil (e.g. linseed oil or tung oil) it's unlikely it would be weeping like this after only a couple of days post-application except in unusual circumstances.
But there are a few rule-in, rule-out things you can look at to try to narrow it down.
A good starting point is the smell. If it's mineral oil there could be no odour from the oil at all and if it has a characteristic smell it's likely something else. Both linseed or tung oils have a smell but they're impossible to describe really (in chemical circles they're said to have a "characteristic" odour, which means you know it when you know it!) but very broadly linseed oil has a sort of 'nutty' odour. Tung oil is sometimes said to have an unpleasant or disagreeable odour (this fades upon drying) but that's also said of linseed oil which many people including me really like the smell of!
Another thing is the colour of the oil, mineral oil being clear as water while almost all oils of vegetable origin are some type of yellow or orange. Linseed oil can be particularly dark (orange-brown) and tung oil is a lighter version of the same sort of colour.
Pretty much impossible to tell for sure without some form of spectrographic testing, but 99.9999% likely that it's safe.
It is probably either mineral oil (ie baby oil), or tung oil or danish oil. These are put on wooden items to provide a (small) level of protection and to bring out (or 'pop') the grain to make it look visually more attractive.
That said, normally the oil should fully penetrate, and if it's a drying oil (like eg tung) it should, well, dry.
Probably this is mineral oil, and the maker may have not have bothered wiping off the excess. I would suggest wiping off as much as you can (you could even use acetone here to help), and leaving it for a day or so and seeing if that helps.
Also, I can't identify the wood from the photo, but some woods such as cocobolo or teak are naturally very oily - it's possible it is just natural oil from the wood.