I’m no Ukrainian constitutional lawyer, but taking a plain-English read, I’m skeptical that Article 17 explicitly prohibits use of the military to prevent people from leaving.

https://ccu.gov.ua/sites/default/files/constitution_2019_eng.pdf

The Armed Forces of Ukraine and other military units shall not be used by anyone to restrict the rights and freedoms of citizens or with the intent to overthrow the constitutional order, subvert the bodies of power or obstruct their activity.

Okay, but is it a right? Well, there’s Article 33:

Everyone who lawfully stays on the territory of Ukraine is guaranteed freedom of movement, free choice of place of residence, and the right to freely leave the territory of Ukraine, with the exception of restrictions established by law.

The “with the exception of restrictions established by law” seems like it’s okay for the Rada to impose restrictions on that right. That is, it’d just prevent the executive portion of the government from doing that without legislative permission. And according to the article, Zelenskyy is asking the legislature to pass this, so it seems kinda like he’s taking the correct route.

The government submitted a new draft of the mobilization law to parliament on Jan. 30, more than two weeks after withdrawing its initial, contentious version.

And I kind of doubt that this was just overlooked.

And registering an account with the government hardly seems like a violation of privacy.

Yeah, but the devil is in the details there. I can’t imagine just having an account being an issue, but maybe there’s some sort of case law that the government can’t require people to notify people of their current location or something, and it might be that this is required.

One more point – the Ukrainian constitution also provides that certain rights can be restricted when martial law is in force, in Article 64:

Article 64

Constitutional human and citizen’s rights and freedoms shall not be restricted, except in cases envisaged by the Constitution of Ukraine.

Under conditions of martial law or a state of emergency, specific restrictions on rights and freedoms may be established with the indication of the period of effect of these restrictions. The rights and freedoms envisaged in Articles 24, 25, 2 7, 28, 29, 40, 47, 51, 52, 55, 56, 57, 58, 59, 60, 61, 62 and 63 of this Constitution shall not be restricted.

Article 17 is not in the “cannot be restricted” list.

Martial law has been declared since about the start of the war:

https://en.wikipedia.org/wiki/Martial_law_in_Ukraine

President Volodymyr Zelenskyy declared martial law on 24 February 2022, in response to the Russian invasion of Ukraine.[26]

On 22 May the Ukrainian parliament extended martial law for another 90 days and automatically renews from that point on.

I would assume that the government is going to maintain martial law for about as long as Ukraine is fighting the war, and as long as that is the case, I believe that Zelenskyy is probably acting in accord with the constitution just on the above point alone.

And while I’m not going to track down the source of all of the privacy objections – several reasons, including treaties, were mentioned – one thing that was referenced was the Ukrainian constitution. The only reference to privacy is in Article 31 (well, and maybe you could count Article 32, for storing data about people), which also is not in the “exempt from being restricted under martial law” list.