Why does the interior paint absorb water and darken when splashed

At minimum, you have to let the paint cure before you can really compare it. It will blend in over time for the reasons bib mentioned.

It might also help to wipe down your walls - smoking, cooking, dirt and grime will discolor them, sometimes noticeably.

Find out how the new paint was applied. If the original paint was applied with a roller, and the patch with a brush, there will always be a noticeable texture difference.

If the patch was not primed before it was painted then this might cause a color difference too.

Really the only way to get a perfectly blended wall is the repaint it.

The standard repainting advice applies:

1. Get matching paint. Ideally, the same paint as was used on the product. Failing that, buy a small container of acrylic interior-exterior paint in a matching color and flatness/glossiness from a paint store. Acrylic paint can be cleaned up with mild detergent and water.
2. Choose a day with good conditions for paint and primer to "level" and dry. Temperature of 50°F - 95°F (10°C - 35°C), not very humid, not rainy. Work in a well-ventilated area that does not have dust in the air.
3. Test the paint next to the rubbed off spots with a fingernail. If you can peel it off with a fingernail, scrape it off. If the object might have lead paint on it, follow best practices for cleaning up lead paint dust.
4. Use painter's tape to mask around where you will touch up. Protect any areas that the paint might drip or splatter onto. If you are really messy, wear eye protection while painting.
5. Roughen the surface with a sanding block or similar tool. Optionally wear eye protection and a dust mask while sanding. If the object might have lead paint on it, follow best practices for cleaning up lead paint dust.
6. Prime, using a paintbrush and water-cleanup primer. Clean your priming paintbrush afterwards.
7. Wait the amount of time recommended by the primer manufacturer. If the workpiece is warm, this should be about an hour.
8. Repaint, using a paintbrush. Clean your paintbrush afterwards.
9. Optionally, apply a second coat of paint after the amount of time the paint manufacturer recommends between coats. This might be a few hours, or it might be a day. Clean your paintbrush afterwards.
10. Remove the painters' tape, and finish cleaning up.
11. Wait a week before doing anything to the object that might rub off the paint. The paint manufacturer's instructions will mention a "cure time"; this is longer than the time until the paint is "dry to the touch".