RAW (Rules As Written), Sentient Magic Items cannot lose concentration from Constitution Saving Throws
Sentient magic items are often formidable and powerful, and influence the world with wills of their own. However, for the purposes of the rules, there is some strong consensus that sentient Magical items are objects, and not creatures. There is also textual support for this interpretation: the DMG gives guidance for creating a sentient magic item, and states (DMG, p. 216, bold added):
Use the information on creating NPCs in chapter 4 to develop a sentient item's mannerisms, personality, traits, ideals, bonds, and flaws... ignore or adapt any result that doesn't make sense for an inanimate object.
This at least strongly implies that even though they are intelligent and capable of thought (and sometimes action) outside of their user, a sentient magical item is considered an object for the purposes of the rules. As such, they are subject to the rules for objects, including the following from the Player's Handbook (p. 185, bold added)
Objects always fail Strength and Dexterity saving throws, and they are immune to effects that require other saves.
They are subject to the rules on spellcasting in general, so they can lose concentration by dying or becoming incapacitated, having the spell's duration end, or by casting another concentration spell (PHB, p. 203-204). But the most common way of losing concentration for creatures is to fail a Constitution Saving Throw from either taking damage or environmental factors (ibid). And since sentient magical items are "immune to effects that require" Constitution saving throws, they will never lose concentration from (for example) taking damage, unless that damage destroys them or (if such a thing is possible) renders them unconscious.
It's also worth noting that many features of the game that impose the incapacitated status (e.g. the spell Tasha's Hideous Laughter, PHB p. 280) do so by targeting creatures, rather than objects. So sentient magical items are also difficult to incapacitate (or stun, or paralyze, etc.).
A DM could overrule any of these interpretations or rules, of course: they might declare that sentient magic items are creatures at their tables. But it's worth noting that the rules as written leave a lot of how that would work ambiguous (the best example is the one you mentioned: they don't have constitution scores, so how would they roll a Constitution save?). If you go by the rules as they are presented, it seems to me that there are very few things in the game that could cause a sentient magic item to lose concentration, other than killing it or making it concentrate on something else.