No, you couldn't design your own units, but let's talk about balance.
Balance includes things like how long it takes this unit to make it to the field, what its vulnerabilities are, what its strengths are (attack), and its cost. If, in the course of 50 turns, I can put out 2 stacks of 12 units, armed with good attacks (which we'll define as an attack 30), those units can put out 360 attack in 50 turns, or a strength gain of +7.2 atk/turn. If I build a monster that takes 50 turns (I know we don't, but stick with me here), if it has a 360 attack, and the hit points to match it, it's actually a better thing to have than the stack, because it will continue to put out a 360 attack all the way through its life on the battlefield, whereas the stack of units, as figures are killed, starts to put out less and less damage, making it a progressively weaker unit as it is worn down.
What balances that is that the stack averages it's 360 attack over a wider range of output. It does 12 30 strength attacks, so the probability of it doing SOME damage each turn is pretty good. If the monster only gets a single, 360 point attack with the same odds of hitting as the other unit, it may miss a lot, or just have poor luck.
Solutions to balance this would be to have the monster have better probabilities of hitting, or be able to attack faster, or (like MOM) have damage from the attack be able to spread over multiple units (so that the monster can kill, say 6 units at a whack on a good roll).
Really, to balance combat, at least to me, is a combination of hit probabilities (dexterity and/or intelligence), damage on hit (strength), evasion (dexterity), hit points (constitution).
There are solutions here, we just have to have enough variables to move things around. With the queued battle occuring in EFE like in HOMM, balance could be something as simple as "with more units in the stack, the speed of the stack goes down (gets fewer attacks per turn)". You can bring more pew-pew, but you ainna gonna fire that often, so it might be better to have fewer units that can take out some of your pew-pew before you even get a shot off.
Just for completeness' sake, Das, MOM did have same equipment across unit types (bowmen, spearmen, swordsmen, and halberdiers were common to most races). You did not decide how many units in a stack; that was decided for you. All units regenerated on the tactical map, with HP recovery being a percentage based on your total, and it happened independent of where you were on the map (MOM did not have ZOC).