It is two taxes, and the "employer match" is arithmetic rather than statute. Section 3101(a) provides that "there is hereby imposed on the income of every individual a tax equal to 6.2 percent of the wages," and 3101(b)(1) adds 1.45 percent for Medicare. Section 3111(a) then provides, separately, that "there is hereby imposed on every employer an excise tax, with respect to having individuals in his employ, equal to 6.2 percent of the wages ... paid by the employer," with 3111(b) adding the same 1.45 percent. So the employee's liability is a tax on the individual's income and the employer's is an excise tax on the act of employing someone. They land on different taxpayers, they are different kinds of tax, and they are equal in amount by design rather than because one is a share of the other. The distinction stops being academic the moment the two diverge, which they do for the additional Medicare tax.
The additional Medicare tax has no employer half. Section 3101(b)(2) imposes a further 0.9 percent on wages above $250,000 on a joint return, $125,000 for a married person filing separately, and $200,000 in any other case. It sits in Subchapter A, the employee chapter, and section 3111 contains no matching provision. Those thresholds are statutory and are not adjusted for inflation.
The employer collects the employee's tax and is on the hook for it. Section 3102(a) provides that "the tax imposed by section 3101 shall be collected by the employer of the taxpayer, by deducting the amount of the tax from the wages as and when paid." Section 3102(b), headed "Indemnification of employer," then provides that every employer required to deduct the tax "shall be liable for the payment of such tax, and shall be indemnified against the claims and demands of any person for the amount of any such payment." So once the tax has been withheld, the obligation to hand it over is the employer's rather than the employee's, which is also why unpaid withholding is one of the exposures a business entity does not shield its responsible people from.
"Contributions" is a misnomer with a history. The chapter is titled Federal Insurance Contributions Act, and the word contributions carries an implication of a premium paid into an account. Neither operative section supports it. Section 3101 and section 3111 both say "there is hereby imposed ... a tax," the money is collected through the tax system, and what the payments buy is a statutory earnings record rather than a contractual claim on a fund. The insurance framing is real in the sense that the programs pay on events, retirement, disability and death among them, but the payment mechanism is taxation.
What FICA is not. It does not include income tax withholding, which comes from Chapter 24 and is the employee's own income tax paid in installments. It does not include federal or state unemployment tax, which the employer owes on its own account. And it does not apply to a self-employed person, who instead pays self-employment tax under a different chapter, computed differently but funding the same two programs.