Who owns the stock in a 401(k)?

(This question is no different from "what happens if my stock broker goes bankrupt", which has been asked and answered multiple times.)

The SEC swoops in and arranges another company to take over administration of the plan (all the assets owned by you and your coworkers). This is essentially no different than when your company changes plan administration.

If there had been some fraud or malfeasance and your shares actually disappeared, then the SIPC would insure you up to $250K.

Just to be clear, X in your example is your employer, and Y is the investment broker.

The answer to your question is that the stock really does belong to you, not your employer or the broker. The assets inside your account are your property, and you control any voting rights from the stock in the account.

If the broker goes bankrupt, you still get to keep the stocks, as they are insured by SIPC, an agency that steps in when a brokerage has financial difficulty to protect investor assets.

You (the individual) own the stock in a 401k purchased with your own contributions to the account. You also own the stocks purchased with the employer's (usually matching) contributions, unless there is a vesting schedule and you have not met the requirements to be fully vested yet.

If "X" (your employer) goes under, then the investment company ("Y") contacts you directly to close out the 401k. They often will automatically transfer the entire account to an IRA.

If "Y" goes under, then your account is transferred to a new administrator (per SEC guidance). I would expect your employer to handle most of the transition though.

The title of the stock is at the investment company, company Y, in an account held in trust for your benefit.

You are protected from various scenarios described by the SIPC and the Securities and Exchange Commission for the case of the investment Company Y failing, and because this is a retirement account, also by provisions in the Employee Retirement Income Security Act of 1974, or ERISA, regulated by the Department of Labor. As an aside a lesser-known key benefit derived from ERISA is that assets in retirement accounts are protected from creditors and bankruptcy with the notable exception of the IRS and domestic relations orders (i.e. a judge can order a split in a divorce case or for child support).

If X goes belly-up, does anything at all happen?

From the Department of Labor ERISA fact sheet:

What if your employer goes bankrupt? Generally, your retirement assets should not be at risk if your employer declares bankruptcy. Federal law requires that retirement plans fund promised benefits adequately and keep plan assets separate from the employer's business assets. The funds must be held in trust or invested in an insurance contract. The employers' creditors cannot make a claim on retirement plan funds. However, it is a good idea to confirm that any contributions your employer deducts from your paycheck are forwarded to the plan's trust or insurance contract in a timely manner. Significant business events such as bankruptcies, mergers, and acquisitions can result in employers abandoning their individual account plans (e.g., 401(k) plans), leaving no plan fiduciary to manage it. In this situation, participants often have great difficulty in accessing the benefits they have earned and have no one to contact with questions. Custodians such as banks, insurers, and mutual fund companies are left holding the assets of these plans but do not have the authority to terminate the plans and distribute the assets. In response, the Department of Labor issued rules to create a voluntary process for the custodian to wind up the plan's business so that benefit distributions can be made and the plan terminated. Information about this program can be found on the Department of Labor's Website at dol.gov/agencies/ebsa.

I give money to [my employer]

You never actually did this. They allocated part of your pay to the 401(k) provider.

Who do those stocks belong to?

They belong to the entity set up for the fund, but you own a portion of that fund.

Generally, retirement accounts won't hold equities directly, but mutual funds. Fund houses set up separate, standalone legal entities for each of their funds, so even if the fund house goes under, none of the assets were under their name.

https://www.kiplinger.com/article/investing/t041-c001-s001-how-mutual-fund-assets-are-protected.html