Can we use $y * \operatorname{sgn}(\hat y)$ as a loss function in linear regression? Can we use $y * \operatorname{sgn}(\hat y)$ as a loss function in linear regression where $\hat y$ is the prediction and $y$ is the target value, or is there any other loss function close to this which can be used?

In regression, where you want to approximate the measured points $(x_i, y_i)$ with those from your estimator $(x_i, \hat y_i)$, a loss function should be a function involving some kind of distance between $y_i$ and $\hat y_i$, like e.g. the norm of the residual $\hat y_i - y_i$.
In your case, you have only one $y$-value in your loss, so, unless something is missing, this would not compare the measured data with the estimated data, i.e. it doesn't measure any loss.
So, the answer is: No, this is not a sensible loss function.
