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?

Let's exemplify with a bivariate normal distribution with $\vec{\mu} = \vec{0}$ and $\Sigma = \begin{bmatrix}1 & -0.9 \\ -0.9 & 1 \end{bmatrix}$.
Import libraries.
import numpy as np
import matplotlib.pyplot as plt
import tensorflow as tf

Generate data set.
cov = [[1,-0.9],
       [-0.9,1]]
x,y = np.random.multivariate_normal(
    [0,0],
    cov=cov,
    size=1000).T
x = tf.reshape(tf.constant(x), (-1,1))
y = tf.reshape(tf.constant(y), (-1,1))

And let's define your loss function.
def sign_product_loss(y_true, y_pred):
    return tf.matmul(y_true, tf.math.sign(y_pred), transpose_a=True)

Define a simple linear model (slope and intercept as parameters).
model = tf.keras.Sequential(
    [
        tf.keras.Input(
            shape=(1,)
            ),
        tf.keras.layers.Dense(
            units=1,
            activation=None
            )
        ]
    )

For training, I selected the Nadam (Nesterov-accelerated adaptive moments) just b/c I find it a reliable default for toy problems. Other optimizers would also work excellently on this simple linear regression problem. In Tensorflow we must compile the model.
loss = sign_product_loss
opt = tf.keras.optimizers.Nadam()

model.compile(
    optimizer=opt,
    loss=loss
    )

Now train the model.
history = model.fit(x, y, epochs=100)

If we look at the loss history, we can see that the model did not obviously trend toward better or worse fit.

And plotting the predictions and the original data together, we can see that the resulting fit was not acceptable.


One way to study an objective function is to plot the contours of its surface over its input. See whuber's answer for an explanation. As a supplement to that answer, here is some quick Python code to produce filled contour plots.
import matplotlib.pyplot as plt
import numpy as np


def loss(x,y):
    return y * np.sign(x)

X, Y = np.meshgrid(*[np.linspace(-1,1,200)]*2)

Z = loss(X,Y)


fig, axes = plt.subplots()
g = axes.contourf(X, Y, Z, cmap='rainbow', levels=15)
##g.clabel(fontsize=10, inline=1, fmt='%.1f', colors='k') # Labels if you want them
axes.plot(*[[-1,1]]*2, color='k')
axes.text(-0.05, 0.05, '$y = x$', rotation=45)
axes.set_xlabel('x')
axes.set_ylabel('y')
axes.set_title('$y \\times \\operatorname{sign}(\\hat y)$')
axes.set_aspect('equal')
plt.tight_layout()
plt.show()

The above can be adjusted as desired, but currently outputs:

