Is it normal that the tyre fits effortless the rim? I've been constantly having problems with my wheel going flat. I couldn't find anything on the rim but I changed the rim tape and I bought a new tyre.

The old tyre was a bit worn. It was also fitting the rim too easy, something I was attributing to the fact that it was old and also that I've been changing the tube a lot.
Nevertheless, now I noticed that the new tyre also goes a bit too easy, as in, I didn't even have to use any tool to put it.

I am sure they are the right size and I don't see any evident sign in the rim to make me think it is not usable anymore, although it is a bit hit here and there but nothing particular as to make anyone think it shouldn't work fine.

Is this normal? Should I check something else?

EDIT: 
The  tyre size is 37-622. It is a city (Grandma/Grandpa-like bike).
Not all tires are hard to put on (depends on type) and rims also have a variation in size (both between different types of rims and sample to sample variation within one model). In fact, I'd say many tires for urban and mountain bikes can be mounted without tools if you know how to. There are obviously counterexamples that are notorious for being tight.

Unless you can get the tire off easily and without tools too, you should be fine.