Before I jump into my today's article I would like to break the topic and talk about fire and matter singly before entering the topic of the day.
First, what is fire? It's a phenomenon of combustion manipulation in form of light and heat as a result of oxidation ( addition of oxygen and removal of hydrogen and the chemical reaction producing the flame is exothermic. 
Matter is any substance that has mas and occupy space by having volume. It's also the atom, molecule and ions which is used in the manufacture of physical substances. They exist commonly in three states solid, liquid, and gas. But, science has outgrown the three major phases, now matter has 2 other states this are plasma, bose-einstein condensates. Thus we are going to talk on the three major states.(solid liquid and gas ).
SOLID: they have a definite shape,volume. The molecules transition around their main position, they can also change states from solid to liquid(melting)and from liquid back to solid which is known as freezing. Solid can also change state directly to gaseous state which is known as (sublimation) but can't move directly from gaseous state to solid state without going through liquid.
LIQUID: it's an incompressible liquid that takes the shape or its container but, retains a constant volume and no fixed shape. Liquid changes state varyingly from liquid to solid (freezing) ,solid to liquid(melting) , liquid to gaseous ( evaporation), and gaseous back to liquid (condensation).
GAS: they spread apart to fill containers, unlike liquid and solid. They have no fixed shape or volume and low density. When heated they get excited and bounce around its containers.
Thus, looking for an explanation for why fire is a matter is somewhat absurd in that absurd. Science study and explain what we do not understand. For a certain substance to be called matter it must possess weight(mass) and occupy space.
Fire can't be contained indefinitely as long as reactant particles are present(oxygen,fuel,orsubstance that causes effective burning).
More so, fire comprises of various gases leading to combustion which eventually librates various gases ,chemical reactiin, light and heat. Personally I think fire is a matter because it could be seen which means it occupies space and it's a mixture of gases which has a certain mass or volume.
Drop your comment if you think otherwise thanks for reading.