5 years isn't bad if the doors get any direct sunlight. If the door is getting beat on by the sun for a good part of the day, you should expect to re-apply varnish every 2-3 years. The sun wears it away and once the varnish is gone, then the weather can start eating away at the stain layer. This is why we try and maintain the varnish layer. The difference between a paint and a stain is that the paint will peel and the stain will not. Usually, with stain, you can just sand away the flaking paint and re-coat. I am not sure why he used a deck stain though, but it probably doesn't matter much. Deck stains have a bit more durability and probably more sheen than a siding stain because they are formulated for foot traffic. I would just continue using whatever he used instead of painting the door.

What do you mean by stained? Zinsser makes a waterproofing paint that is good for painting concrete block.

The traditional answer is that an oil based primer should be used because tannin is water soluble. However, some of the new acrylic, water based primers, such as Kilz Premium, will stop tannin bleed, provided that 18 hours of cure time is given before top coating with a water based paint. Some of the new "paint and primer" paints are also stain blocking. Two coats of the paint itself will stop tannin bleed, but again, only after an overnight cure time before applying the finish coat. And yes, water based paints can be applied over a thoroughly dried oil based primer.

this is your photo i am trying to replicate, the french blue island. I painted the blue now I just need to know how and with what the staining was done with.