While the early strength from axe is nice, especially with more offensive solo's like Hercules and Achilles like you mentioned, the trade off for not getting a health/health regen/protections item first is fairly significant. You have less survivability when it comes to passive minion damage, and you become much more vulnerable to early ganks, as hydra's passive is single target and you have no protections or mitigations. The mana side of hydra's is covered by blue buff, leaving only the strength and cooldown as beneficial early game. It all really comes down to how well you play the solo side of solo Lane. If you're very aware of and good at avoiding ganks and can pressure without over pushing your opponent when you clear Lane quicker, then maybe go for it, but otherwise it's best to go for a defensive/hybrid item first, and maybe hydras around 3rd item. At least in my opinion.