Is it normal to have two (or more) apps with the same codebase and database in Google Play? I installed two Google Play applications and saw, that they are actually the same app (completely same functionality and database) with only different splash screens. Isn't it against the rules?
It's not normal but there's nothing to forbid a developer releasing multiple variants of the same product with different branding. Some companies even develop white box apps, which their customers add their branding to and publish, the same way that one off-brand gadget or home appliance is sold by different companies under different names.

It's only forbidden if one developer violates copyright by publishing another developer's app without their consent; for example, if I decompile the APK of Angry Birds, replace the name and splash screens with "Bangin' Avians", and publish it on Google Play, then I'm violating Rovio's copyright on their product.