Some countries have a document they call a "seaman's passport" in English (such as Dominica, which Ana has cited), but very few do, and Uruguay does not seem to be one of them. And when they do it isn't actually a passport; the term refers to a seafarer's identity document (commonly known among English-speaking mariners as a "seaman's book"), which is the "libreta de tripulante" referred to in this same text. This is true of Bulgaria, for example:

"The Bulgarian identity document issued for the purposes of Convention 108 of the International Labour Organisation is the Seaman’s Passport."
http://www.starnav.eu/seaman’s-passport

See:
"The provisions in the ILO’s 1958 Convention No. 108 (ILO C108 or “ILO 108”) created an internationally-recognized identity document for crew traveling aboard seafaring vessels [...] The ILO and the maritime industry refer to this document as a Seafarer’s Identity Document (“SID”), also commonly referred to as a “Seaman’s Book”."
https://www.maritime-executive.com/article/offshore-and-mari...

In any case, the term here is not "pasaporte de marino", which doesn't exist in Uruguay, but "pasaporte con constancia de marino": in other words, just an ordinary passport which states that the holder is a seafarer.

I've never seen a Uruguayan passport so I don't know whether they routinely state the holder's profession. I've a feeling they don't any more, but the following suggests that they did in 2008:

"Natalia ha entrado tres veces a España en épocas recientes, todas con procedimientos tan truchos como impecables por su eficacia. Tiene 35 años y su pasaporte uruguayo dice Profesión: diseñadora. [...]"
http://www.lr21.com.uy/politica/317735-consejos-infalibles-p...

(UK passports don't state the holder's profession either, but they used to. I still have my first passport, issued when I was 12, and it says "Profession: Schoolboy".)

Anyway, as I see Phil has commented above, the preferred official term is "seafarer" rather than "seaman".

--------------------------------------------------
Note added at 6 hrs (2018-12-14 20:20:39 GMT)
--------------------------------------------------

Just to clarify the point Phil has raised in his comment, he's right that this doesn't necessarily refer to a Uruguayan passport. And it is true that there are a very few countries that issue a special kind of passport for seafarers. As far as I can discover they are China, Egypt, Korea (North and South), Libya, Syria, Turkmenistan, United Arab Emirates and Vietnam. I am not sure about Bulgaria and Russia, and Panama seems to be planning to introduce one. But in the vast majority of countries there is no such thing. Other instances of this phrase seem to refer to the "seaman's book".

It is inherently unlikely that this Uruguayan law is making a special provision for these exceptional cases, and if it were referring to a special seafarer's passport held by people from those countries it would have said "pasaporte de marinero". "Pasaporte con la constancia de marino" simply means a passport that states that the holder is a seafarer (or that is accompanied by documentary evidence that the holder is a seaferer). Of course, a seafarer's or seaman's passport held by a sailor from Vietnam, for example, would meet this requirement, but so would an ordinary passport from any country that stated that the holder's profession is seafarer, and in the vast majority of cases it would be the latter. So I think the translation must be expressed in a way that covers these cases.