I would argue that Telemachus does not quite show himself to be equal to Odysseus, but he clearly shows that he is obedient to his father.
Telemachus almost shows himself to be equal to his father because he is going to be able to string the bow. However, it is only on his fourth try that he looks like he is going to be strong enough to bend the bow and get the string in place. Odysseus, by contrast, is able to string the bow very easily. So Telemachus is closer to being equal to his father than the suitors are, but he is clearly not quite equal.
It is clear, though, that he obeys his father. When Telemachus is about to succeed in stringing the bow, he sees his disguised father give him a sign. When Telemachus sees the sign, he immediately gives up trying to string the bow. This shows that he obeys his father even if it makes Telemachus himself look bad.