Can I merge multiple linear regressions into one regression? I have a dataset of 48,000 records
These records are divided equally into 4 groups
Blue Group, Yellow Group, Green Group, and Red Group
Each one of these groups has 12,000 records
My study has resulted in 4 regression models

Model 1 equation was $a_1 x + b_1$
Model 1 equation was $a_2 x + b_2$
Model 1 equation was $a_3 x + b_3$
Model 1 equation was $a_4 x + b_4$
The thing is that the dataset is not available anymore and I want the regression model for all dataset points
Can I get the average of all 4 models to get the regression for all data points?
So it will be $((a_1 + a_2 + a_3 + a_4)/4)  x + (b_1 + b_2 + b_3 + b_4)/4$
is this model the right model for all data points?

The answer is no.
You were learning the coefficient independently without considering the 4 features together.
Let me use a simple example to explain.
Assume the real regression result is: $10*x_1 -1*x_2 = y$
Assume in your samples $x_1, x_2$ have a positive relationship with $y$. Meaning when $x$ increases $y$ increases too. This is not too hard to do by choosing the samples, e.g.
1, 2 -> 8,
2, 3 -> 17,
3, 4 -> 26
...

Learning coefficient of $x_1, x_2$ separately will always give positive coefficients $a_1, a_2$.
Never will it learn $-1$.
