Is it mathematically wrong to use units instead of words/parameters/names in equations?

A specific parameter might correspond to a specific (SI) unit, but not all units correspond to a specific parameter!

Kinetic energy is

$$\begin{align} K&=\frac{1}{2} mv^2 \\ [\text{Joules}]&=\frac{1}{2}[ \text{kilograms}\times\text{meters}^2/\text{seconds}^2] \end{align}$$

We also have gravitational potential energy:

$$\begin{align}U&=mgh \\ [\text{Joules}] &= [\text{kilograms} \times(\text{meters} / \text{seconds}^2) \times \text{meters}]\\ &= [\text{kilograms} \times \text{meters}^2 / \text{seconds}^2] \end{align}$$

So, is Joules both $\frac{1}{2} \text{kilograms}\times\text{meters}^2/\text{seconds}^2$ and $ \text{kilograms}\times\text{meters}^2/\text{seconds}^2$ at the same time? If you have a value in Joules and you need to find the number of kilograms, then how would you go backwards? How would you do the algebra?

You could start from any of these unit-formulations, and you would get difference answers for the number of kilograms. The answer is not unique seen from the units since the original formula could have contained unit-less parameters.

The problem is that there are many kinds of energy with the same unit. In general, parameters have unique units, but units don't belong to unique parameters. You cannot go "backwards" from the unit formulation of a formula.

For simple equations, the two might be equivalent. Certainly, dimensionally an equation must always be correct. But there are plenty of situations where the units may not obviously reflect a particular quantity; and clarity of communication improves understanding.

Take electrostatics. If I say "1 Volt" you know what I mean; an electric field is "Volts per meter" - still OK. But what if I used a quantity with units $\rm{kg~m^2~s^{-3}A^{-1}}$? Would you know if that was a voltage, or an electric field?

Conventions develop because when everybody "speaks the same language" you spend less time decoding, and more time thinking about the underlying physics.

PS - it's voltage.

Writing equations using only units would not work at all for dimensionless equations. For example the Snell's law $$n_1 \sin\theta_1=n_2 \sin\theta_2.$$ You would also lose many of the dimensionless (but usefull) parameters in physics such as the Lorentz factor $$\gamma=\frac{1}{\sqrt{1-v^2/c^2}}.$$

Also consider equations whose all variables have the same units. For instance the (truly fundamental) first law of thermodynamics, $$\Delta U=Q-W.$$ It would be meaningless if we write it only in terms of units.