If you want males to give birth, they need genitals that can give birth. That's why male seahorses don't have external
penises (and why they have vagina-like openings).
If you want two transfers of genetic information, do it safely. You ask for two, but seahorses only have one.
The detailed version
Seahorses have a one-way transfer - females inseminate the males. Going back and forth twice (male->female->male) as you suggest increases the risk of losing the egg, so in nature, there is usually just one transfer.
Conversely, if the male gives birth, he will have an opening large enough to do so. This is why male seahorses have vagina-like openings and not external penises - if you're taking the "male birth" part it is important to take the genitals that allow male birth.
I'm going to try to find a balance between "identical" genitals and males that give birth, but obviously you cannot pass a baby through a penis. These facts are hard to change, so "identical" genitals must be stretched.
Safe transfer of fertilized eggs from female to male
This is possible, but even seahorses don't risk it; it's easiest to transfer genetic material once. Consider one of the following ways to make the transfer safely.
- Males could "suction" the resulting mixture with the penis.
- If you want human-like birth rates (on average, one child) then this is not safe; it's too risky to lose the only egg there is.
- If you opt for more than one offspring each time (seahorses have thousands of offspring, for instance) the risk may be negated; why does it matter if you lose a few eggs when most of them made it?
- The female could gather the fluid containing the egg, and condense it (or drain it into a nice pod), then eject the egg in this substance / object back into the male through her own process (orgasm? round 2? who knows.).
- This has the added bonus that you can complicate births further. Males can receive the eggs of other males.
- You could ditch eggs during this transfer and bring them in at the last step. Males could ejaculate something similar to sperm into the female, who would edit her own DNA into as many sperm as possible. This final fluid would be siphoned or ejected into the male, and the first one to reach the lining of the male's uterus would begin to develop - similarly to the "first sperm wins the race" idea.
Safe male birth
There are three main ways to do this - to have a vaginal opening, to lay eggs, or to be severely mutilated (die?) in the process of childbirth.
- The "vaginal opening" idea sounds most promising. You describe males with uteruses (uteri? what is the plural of uterus?) so it seems like this would be the case
- You can choose to have the penis expand, just as mammalian vaginas contract and grow larger, to give birth, although I'm guessing it would be extremely uncomfortable. It may render the penis useless.
- Alternatively create a new opening adjacent to the uterus; this is safe, comfortable, and can be out of the way.
- Egg-laying via the penis (oviposition) will allow you to keep genitals as they are
- This would defy the mammalian characteristic of giving live birth, but it would require the least physical change
- Mutilation during childbirth (no organs suited to birth, so the offspring crawl out) is possible, and may even occur in nature. This may be too gruesome of a path.
Note: I don't know if anything I've said here is realistic from an evolutionary standpoint. The idea of male to female to male transfer is already stretching it past what we've seen on Earth.