It doesn't sound genuine to me as a Buddhist myself following one of the Tibetan traditions. There are many quotes that are falsely attributed to the Dalai Lama, just as for most famous public figures.

The only Buddhists who are not supposed to handle money are the Buddhist monks and nuns, and only if they have taken full ordination, not the novice vows - and that's regarded by Tibetans as a minor rule, not one of the essential core rules. So monks will often buy things in practice in our society because to not do so would inconvenience others, mean they have to have someone to do all their purchases for them. It made sense at the time of the Buddha with a tradition of homeless spiritual seekers supported by donations of food etc and begging, and highly regarded for their spiritual path they have chosen.

Therevadhan monks are stricter, but Tibetan monks and nuns have this approach where they consider that compassion overrules all the minor rules of conduct. The reason for this difference of interpretation is that Buddha said that some of the rules were minor ones that could be ignored, but he never clarified which were the major and which the minor ones - according to the sutras he was never asked, his disciple who was responsible for remembering the rules of conduct never thought to ask him. So it is open to different traditions to have different intepretations here, and they do.

There is nothing at all in Buddhism about lay people not handling money, at all. It's normal for lay Buddhists to work, earn money, pay for things, save up, etc etc. Buddhist teachings do encourage generosity. But not giving away all your possessions - that's a Westerners misconception of what it is about that some have.

So, I don't see how a Tibetan monk could ever say this.

So, reads to me like something composed by a Westerner who has this misconception about Buddhism - or else, as is very common, just a quote from someone else, which has been misattributed to the Dalai Lama.