Poisson regression with continuous data [duplicate]

I have a time series of floating numbers, say,

0.1, 0.5, 1.1, 0.6, 2.0, 1.4, 0.4

Now, I would like to model this series with Poisson regression, since the numbers, even though they seem continuous, are in fact from a specific range/set of possible values from 0.1 to 3 in 0.1 steps: you can have 0.1 and 0.2, but not 0.15. To me it does not seem like very much different from having counts.

I used this transformation: all floats have at most 1 digit of precision limited by the equipment that makes measurements, so I simply multiply them all by 10, and get the transformed time series output of

1, 5, 11, 6, 2, 14, 4

I tried running Poisson regression on it, and frankly speaking it does as well as it did for similar quantities of counts.

What are the pitfalls of this approach? Is there any literature doing something similar?

My biggest concern is the relation of mean-to-variance in Poisson distribution. By multiplying the time series by 10, I automatically change the variance/expectation of Poisson, so the distribution changes. However, I think that that scaling does not actually influence the point prediction, only variance, since I can easily get the inverse transform of the prediction, and my practice showed it is pretty good.

How can this be justified/explained?