This question has been the topic for great discussion; in providing research for my answer, I even found a fair few discussion posts on Reddit, though I can not seem to find them, now.; however there is evidence of regular animals existing in the Pokémon universe, where the lack there of appears to be the most popular argument for Pokémon being animals.
At the end of the day, Pokémon are monsters; after all, Pokémon is a portmanteau of "pocket" and "monster". Some of them are based off animals, and thus act like and can fill the roles of particular animals; others are based off plants, or even inanimate objects. Whether they are actually considered animals is an entirely different story, and there is not enough conclusive evidence.
This question appears to correlate the MRS GREN acronymn with being an animal. In actual fact, the MRS GREN acronymn does not specifically identify an animal; it identifies a living organism. We can say without doubt that they are living organisms. Let's take a closer look at the MRS GREN definition of a living organism.
Pokémon definitely move. They would not be the source of entertainment they are if they just sat there all day. Some move more than others, while some barely move at all, but there is always some degree of movement capabilities.
We do not really see much in the way of breathing, but every now and again, we see a Pokémon that is short of breath (often being out of breath after some overly physical action such as running or taking a severe beating). You do not get short of breath if you do not need to breathe in the first place.
We often see examples of physical sensitivity in Pokémon, usually after taking a big hit or being defeated in battle. We also know that they have a physical endurance level (i.e. they "take damage"), because they can be defeated in battle, and often have to be healed.
Pokémon grow in two ways. They grow in size, and when they reach certain conditions, they evolve and grow into new Pokémon. We often only clearly see the size difference when the Pokémon evolves; however, in some of the games (notably Pokémon Go), you can catch the same Pokémon in a variety of heights and weights.
We know that Pokémon reproduce, because they lay and hatch from eggs. In fact, Pokémon breeding is a commonly employed mechanic used to catch Pokémon of particular rarities and strengths.
OK, so we do not see Pokémon poop. There is a good reason for that; it is a video game and TV series primarily targeted at children. It is one of those things that just go without saying. We do, however, see references to Pokémon excrementing waste, primarily through Pokédex readings:
Darumaka's droppings are hot, so people used to put them in their clothes to keep themselves warm.
- Daramuaka's Pokédex reading in Pokémon X and Pokémon OmegaRuby
A dangerous Ultra Beast, it appears to be eating constantly, but for some reason its droppings have never been found.
- Guzzlord's Pokédex reading in Pokémon Moon
The latter has been used to say that Guzzlord does not excrete; but on the contrary, this goes to show that Pokémon in general do. After all, if they did not, it would not be any mystery why one particular Pokémon's droppings have not been found.
We see Pokémon eat all the time. Ensuring your Pokémon has a nutritious diet is also often an employed mechanic in-game.
In summary, in contrast with part of the question, Pokémon can be considered living organisms; however, there is only discussion and debate about whether or not they are actually considered animals.
After all, Pokémon are fictional organisms. The closest real organisms we have are predominantly animals. If they existed in the real world, they could quite possibly be classed as a completely seperate organism due to the clear distinction of both ability and comprehension. Since that is not the case, we can't really try to classify them with non-fictional constraints.