The criterion for marriage in Islam isn't entitlement to heritage. According to Islam, marriage is of two types: permanent and temporary (mut’ah). The formulas for each have their similarities and differences; this being a result of the reason for why mut’ah was even legislated and permitted in Islam (which was to facilitate marriage for those who needed it but for whatever reason couldn’t practice permanent marriage). Not inheriting from each other is also another natural consequence of the reason behind the whole concept of mut’ah. Of course, let it not remain unmentioned that in temporary marriage, if the couple stipulate that they must inherit from each other, it will be binding, the same way that in permanent marriages, the couple can stipulate that they don’t inherit, which will also be binding.
The criterion for marriage in Islam isn't whether the couple inherit from each other or not. Not inheriting doesn’t have the least effect on the validity of the marriage contract. According to Islam, marriage is of two types: permanent and temporary (mut’ah). Mut’ah denotes a marriage in which some of the rulings of permanent marriage have been overlooked for the purpose of easing the whole marriage process, namely the nafaqah (sustenance) and inheritance between the wife and husband. Therefore, naturally, there will be differences between these two types of marriage, stemming from their respective functions. As for your argument, that holds that if a man and woman are to be husband and wife in Islam, they must inherit from each other, and since mut’ah couples don’t inherit from each other, they aren't truly husband and wife, isn't backed by neither the Quran nor traditions. In the verses of inheritance, all the Quran says is that the wife and husband inherit from each other. At the same time, we know that even if Islam has said that there is inheritance between the husband and wife in a general manner, there can be exceptions, of course, if they are made by authentic sources. Therefore, when the verse speaks of the wife and husband inheriting from each other, when added up with traditions that make an exception for mut’ah, we conclude that what is meant by the verse is the permanent wife and husband.
There are also cases in our hadiths where inheritance has been negated even in permanent marriages. Here are a couple examples:
1- As has been explained in fiqh, if a woman marries a man in a sickness that he eventually dies with, she will not be entitled to any inheritance if the marriage hasn’t been consummated, and of course, the man will also not inherit from her either.
2- If the wife kills her husband, or the husband kills his wife, the killer will no longer be entitled to any of the heritage.
Our question is, in such cases, can one claim that since there is no inheritance, that there is no marriage and that the couple aren't lawful to each other?
There are other examples like these in fiqh too and aren't limited to the ones we mentioned.
In addition, it isn't true that in mut’ah there is absolutely no inheritance at all, because if the couple agree to inherit and stipulate it in the marriage contract, it will be binding.
Muhammad ibn Abi Nasr Bazanti narrates Imam Reza (as) saying: “The mut’ah marriage can be both with and without inheritance; if the two stipulate it, then there will be, and if they don’t, there won't be.”
In closing we find it of dire importance to say that although questions like these come from being precise, nevertheless, we should always remember that in order to be able to derive Islamic law or attribute something to religion, it is necessary to take all dimensions of the issue into consideration and all related sources must be seen first, not to mention to refer to the experts on the matter.
 Tabatabai, Muhammad Hosein, Al-Mizan, vol. 15, pg. 15, the Jame’ul-Tafasir software.
 Such as verse 12 of surah Nisa.
 Mu’izzi Malayeri, Ismail, Jame’u Ahadithil-Shia, vol. 26, pg.99
 Makki, Lum’atul-Damishqiyyah, chapter of inheritance, pg. 427; Book of Civil Law, article 945.
 Jame’u Ahadithil-Shia, vol. 26, pg.99.