a. As his father played tennis, so did he.
b. Just as his father played tennis, so did he.
Do these mean
1. His father played tennis and so did he.
2. He played tennis in the same way his father played tennis.
3. He played tennis because his father did.
4. He played tennis at the same time as his father did.
Sentence (a) can mean (2), (3), or (4). It is context that will make the difference.
Sentence (b) can mean (1). For meaning (2), however, you could say:
(2b) He played tennis just as his father did.