#Short Answer
In the U.S. there is a federal law making unauthorized entry into the country illegal. Does Mexico have similar law?
Yes. Illegal entry into Mexico is illegal but, it is not a crime.
If so, does anyone know of English translations?
An English translation of Mexico's immigration laws is linked below.
#Long Answer
Illegal entry into Mexico is illegal and has legally authorized consequences, although it is not a crime.
Civil detention of unlawful entrants into Mexico is authorized by Articles 99-105 of Mexico's Migration Act of 2011.
Deportation is authorized by Articles 114-125.
There are civil fines for violating immigration laws in Mexico based on a multiple of the minimum wage, ranging from 20 days of minimum wage to 10,000 days of minimum wage, depending upon the offense and its severity (one day's minimum wage is 88.36 pesos). Articles 143-158. So fines range from 1767.2 pesos to 883,600 pesos for immigration violations. Today's exchange rate is 19.4 pesos to $1. So, these fines range from $91 USD to $45,546 USD.
Immigration crimes are found at Articles 159-162. According to the linked translation, these crimes are as follows:
Article 159. A term of eight to 16 years in prison and a fine ranging from 5,000 to 15,000 days of the general minimum wage in effect in the Federal District will be imposed upon an individual who:
I. Traffics one or more individuals in order to enter another country without the corresponding documentation and for the purpose of directly or indirectly obtaining a profit;
II. Introduces one or more foreigners into Mexico without the corresponding documentation and for the purpose of directly or indirectly obtaining a profit; or
III. Lodges or transports one or more foreigners in or through Mexico for the purpose of avoiding migratory inspection and in order to directly or indirectly obtain a profit.
In order to actualize the crime provided for in this article, it will be necessary to
demonstrate that the offender's intent is to obtain an indisputable, current, or impending economic benefit in cash or in kind.
Penalties will not be imposed upon individuals of renowned moral rectitude who, for strictly humanitarian reasons and without seeking any benefit, assist an individual who has entered the country in an unlawful manner, even when such individuals [of renowned moral rectitude] receive donations or resources for continuing their humanitarian work.
Article 160. The penalties indicated in the foregoing article will be doubled when the aforementioned conducts are carried out:
I. With respect to children and adolescents, or when a child or adolescent who is not able to comprehend the significance of the action is induced to, motivated to, assisted in, or obligated to conduct any of the behaviors described in the previous article;
II. Under conditions or using means that place or could place health, integrity, safety, or life at risk or that allow for inhumane or degrading treatment of the individuals affected by such conduct; or
III. When the perpetrator or orchestrator is a public servant.
Article 161. A public servant who assists, conceals, or prompts any person to violate the provisions of this Law in order to obtain a direct or indirect profit in cash or in kind will be sentenced to a term of four to eight years in prison and a fine ranging from 500 to 1,000 days of the general minimum wage in effect in the Federal District.
Article 162. In regard to the crimes indicated in this Law, the Federal Office of the Public Prosecutor will exercise the criminal action on its own initiative. The Institute is obligated to provide the Federal Office of the Public Prosecutor with all elements necessary for prosecuting these crimes.
So basically, the only immigration crimes in Mexico are human trafficking and immigration related bribery. Illegal entry into Mexico, per se, even if it is intentional, while it is illegal and may justify civil immigration detention, deportation and/or a civil fine, is not a crime.
Of course, one could conceivably commit other crimes in connection with the immigration process (e.g. assaulting a government official, forging official documents), but illegal entry into Mexico, per se is never a crime, unlike the situation in U.S. immigration law, where illegal entry is both illegal and also a crime in many (but not all) circumstances.