Multiple linear regression for multi-dimensional input and output?
You are asking about multioutput regression. The class you talked about sklearn.linear_model.LinearRegression supports this out of the box.
import numpy as np
from sklearn.linear_model import LinearRegression

# features
A = 10
# number of values to predict
B = 15
# number of rows in dataset
m = 100

x = np.ones((m, A))
y = np.ones((m, B))

model = LinearRegression()
model.fit(x, y)

sklearn.linear_model.LinearRegression actually just creates B models. However it optimises calculations using vectorisation.
It actually is exactly the same as a fully connected layer in a neural network which has no activation function.
You can read more about it here:
https://machinelearningmastery.com/multi-output-regression-models-with-python/