Let's say in this hypothetical scenario a man wanted to be married to multiple different women in the United States in the State of Tennessee.
Would it be legal for the man to have a marriage license with his first wife and then have private religious wedding ceremonies for the other women but not have a marriage licenses with them?
How is this situation handled when they all file taxes?
A person A who knows that he or she is lawfully married (to B), but conceals this and deceives a third person (C) into thinking that A is not married, and thus induces C to go through the form of marriage with A commits bigamy and in most states, fraud. C would be classed as a "putative spouse", that is as a person who believed in good faith that s/he was lawfully married. A putative spouse gets some of the rights of marriage until the facts become known. The rest of this answer assumes that all parties know what is happening, and that no deception is involved.
No US state currently allows a person to be legally married to more than one other person at the same time. States do not normally restrict what private religious ceremonies people may choose to have. Indeed under the First Amendment's "Free Exercise" clause the possible scope of any such regulation is very limited.
However, as far as the government (state or federal) is concerned, a person has only one legal spouse at a time. A person should not describe himself or herself as being currently married to more than one other person to any government official or agency. No private religious ceremony will give the third person involved any of the legal rights of marriage.
A person attempting to obtain a marriage license or go through a public marriage ceremony while already married to another person my be committing the crime of bigamy. Bigamy is not often prosecuted as a matter of policy, unless a deceived spouse complains. But that is a matter of prosecutorial discretion, not of law.
Under § 39-15-301 of the Tennessee Code subsection (a)(1) or (a)(2), a person who "purports to marry or be married to" another person knowing that at least one of the parties is already lawfully married commits bigamy. This applies if the purported marriage would have been lawful had neither party been in an existing marriage. Going though a marriage to a second spouse with a (falsely obtained) license would violate 39-15-301. A private religious ceremony that did not claim to be a legal secular marriage probably would not, but I cannot be sure of that.
For federal income tax purposes, two people who are legally married should choose either "married, filing jointly" or "married, filing separately" There is no option to file as a three-person marriage, and an attempt to so file will probably cause the return to be rejected, and may cause other problems. (Filing a "frivolous return" is subject to a significant monetary penalty.) A third person who is part of the same household would presumably file as "single". Other tax filings will be handled similarly.
