I'm going to try and answer this "side-on", there are lots of different possible interpretations so I'll try and give a "tour" of them. I will begin with utility, as I think your colleague's claim stems from thinking in these terms.
When talking about anything in terms of utility the same question always rears its head: in the public sphere it arises when people talk of economic models that maximize wealth without regard to human welfare; in medicine it occurs when people discuss the benefits of medical trials vs. the risks involved. The question is "Can we summarise everything that we need to know into a single number, and then use that number to make a decision?"
For the specific case in your question, where it is specifically about (the same) money, it seems completely reasonable to put the punishment and the reward on the same scale. But, if we use a subjective measure of utility, such as happiness, this might be very difficult.
For example, if we associate reward and punishment with some measure of pleasure and pain (as is typical for neuroscience) then it becomes relevant that the physiological pathways involved in learning from reward are separate from those for punishment.. This suggests that rewards and punishments may be very difficult, or perhaps impossible, to put on a single scale. Reward and punishment as understood by neuroscientists are not on the "same axis" and I would expect a neuroscientist to say no to the title question.
But if we make the assumption that it is money that is important, there is still a problem concerning whether the scale it is measured on is "uniform"... For example, if someone has 10,000$ and you take 9,999$ away from them, it is going to have a much bigger impact on their life than giving them 9,999$. A monetary punishment of equal utility to that of the reward would not have the same dollar-value. So, a fictional economist who was simply comparing utility would conclude that a punishment for one is a reward for everyone else, even though they differ in degree. This economist - who works on the principle of only considering objective losses and gains - will not see a qualitative difference between reward and punishment, even though they may differ in quantity.
So, we can safely say that if any difference does exist then it does not lie in the losses or gains. If we stop looking at the consequences of punishment or reward, but look at the reasons for it, we quickly find an answer: Punishment is what you get when break a rule or law, reward is what you get for good things not required by a rule or law.
We find a difference in terms of what is expected of someone in a society - either by individuals or that society at large. It is a normative difference: using this criteria to ask whether something is a punishment or lack of reward depends on not only fairly concrete things like laws but also more tricky things like standards of behaviour, custom and the quirks of human emotion.
It has been my experience that many people reject this kind of distinction due to its subjective, "arbitrary" and culturally variant character. I expect that your colleague would be one of them.