Lasso Regression Assumptions These are the linear regression assumptions:

Linearity: The relationship between X and the mean of Y is linear.
Homoscedasticity: The variance of residual is the same for any value of X.
Independence: Observations are independent of each other.
Normality: For any fixed value of X, Y is normally distributed.

Do they apply to Lasso Regression, too?

I will be a contrarian and say that most assumptions do not apply to LASSO regression.
In the classical linear model, those assumptions are used to show that the OLS estimator is the minimum-variance linear unbiased estimator (Gauss-Markov theorem) and to have correct t-stats, F-stats, and confidence intervals.
In LASSO regression, those are less important. LASSO gives a biased estimator, so any of the Gauss-Markov business for the minimum-variance linear unbiased estimator no longer applies; indeed, LASSO is not even a linear estimator of the coefficients, so Gauss-Markov doubly does not apply. Then the hypothesis testing and confidence intervals would not be of primary concern for a LASSO regression. Indeed, it does not even seem agreed upon how such inference would be performed.
(Linearity still matters, but that’s because OLS and LASSO both estimate the same parameters, which are the coefficients on the features.)
Overall, the typical assumptions are assumed in an OLS linear regression because of nice properties of later inferences that are not particularly important for a situation where LASSO regression would be applied.
