Application of almost any wood oil will darken the wood to some degree and enhance the natural colour tones and character of the timber. The type, age and condition of the wood will all have an impact on the final, oiled colour. Teak, is generally a darker and slightly more red toned wood and any applied oil will highlight this richness.
It will be difficult to maintain the lighter, natural colour with a penetrating product. On a tropical hard wood such as Teak, you could consider Osmo UV Protection Oil Extra Natural (429), however I would strongly recommend a test area first because of the type of wood you are applying to. This product contains a small amount of white pigment that counteracts the darkening effect of the oil. Whilst it works very well on lighter coloured timbers, is can leave a slightly milky hue on darker woods. Some people like this and other don't, so trying the sample sachet is a good place to start.
To use this oil you would need to sand the furniture back to bare wood first, to remove the current finish.
The unevenness in the first finish could be just a result of the wood and the absorbsion of the oil. Wood is not uniform as it grows so some areas can be more dense than others. This means that the oils can be absorbed unevenly meaning that it may sink further into some areas than others. This can usually be rectified by applying another thin coat but it's important to not over apply the oil and to remove any excess from the surface of the wood with a clean, lint-free cloth.