There is no reason you can't paint oak. Be aware that oak has an "open" grain unlike maple or poplar. That means you will see the texture of the wood after it is painted and it will not be perfectly smooth, but it can look great nonetheless. Clean the surface thoroughly with a solution of dishwashing liquid to degrease, sand lightly, and use a high quality paint. If any areas of the finish are sticky, it is either broken-down varnish or accumulated grease and it will need to be removed completely. Good luck to you!

That said, in the after photos, the painted oak does not appear to show any grain. Do you have a closer shot of the painted cabinets so we can see the amount of grain? I would love to paint my oak cabinets white but am still uncertain of the final look.

you can use oil paint, it will make the cabinets look much more smooth, but it is extremely stinky and requires drying for 24 hours before a second coat. Or you can try a high quality primer (like Kilz) to help go over the grain before painting with high quality latex based.

Kitchens wonderful looking!

Those are different doors. They look replaced to me. Slightly larger, different style. Getting pre-painted replacements and then matching the paint for the body? How is that accomplished. End result looks great here. More information on how and how much is needed.