My rear wheel keeps getting side to side wobble issues. Can it be too much pannier weight? It's now already the second time that the rear wheel has issues with side to side wobble and I would like to understand why this is happening.

The first time I had a bike mechanic to tune the true the wheel spokes but then it happened again.

I am not 100% sure but I have the feeling that both times it happened on occasions where I was carrying around 10-12 kilos in a single pannier (on the left side).

Mine is a mountain bike with a rear fitted pannier rack which can support up to 25 Kg of weight.

I know that carrying over 10 kilos on just one side is not ideal, but when I do so is only for a very short distance.

Anyway, I was wondering if that could in any way cause problems to the wheel spokes?
If the rim is aluminum made, truing it after a serious bend can result in fatigue of the material leading to repeated subsequent failures.

I'd try a sturdier replacement wheel instead of repeatedly truing it, especially if it's a job you pay for and not doing it yourself. 

If you are happy with the hub  you can keep it and get just a rim. The cost of building a wheel can be pretty high though, depending on where you live. 

The weight of the pannier should not count too much, unless you are a heavy rider and the total weight on the bike is near the specced limit of the wheel. This is usually the case for low-weight, racing, higher-end wheels (some have a 80-120kg limit) or for cheapo wheels with no specs and poor materials used.

The unbalanced weight should also not account for the issues you are having, given the fact the bike should be able to resist great side forces when turning, especially at high speeds.