Is there any name for regression on response variable between 0 to 1? I know linear regression is on a continuous response variable and logistic regression is on a binary response variable. 
But is there any name for regression on response variable between 0 to 1? Is this approach, where set the objective function to $\text{minimize}~ \|\frac 1 {1+e^{-X\beta}} -y \|_2^2$ valid? and had a name?
 A: If the response variable is between 0 to 1, then you could model using a Beta Regression. The seminal paper is 
Ferrari, S.L.P., and Cribari-Neto, F. (2004). Beta Regression for Modeling Rates and Proportions. Journal of Applied Statistics, 31(7), 799–815.
There is also a R-package available called 'betareg'. An example from the documentation:
library(betareg)
data("GasolineYield")
gy <- betareg(yield ~ batch + temp, data = GasolineYield)
summary(gy)

A: Not sure what the data is you're trying to model, but another option is to use some sort of transformation on your response using a monovariate approach. Something like
$$
y = log(\frac{x - a}{b - x})
$$
where a = lower limit and b = upper limit
This is used a lot in forecasting I believe (check out the article by the always excellent Rob Hyndman: https://robjhyndman.com/hyndsight/forecasting-within-limits/)
A: I believe the generic term is fractional response regression. There are logit, probit, and heteroskedastic probit conditional mean versions.
The standard reference for the logit case is:
Papke, L. E., and J. M. Wooldridge. 1996. "Econometric methods for fractional response variables with an application to 401(k) plan participation rates." Journal of Applied Econometrics 11: 619–632. 
