Some do percentage matching up to a certain percentage of employee's pay. Other do percentage matching up to a certain dollar amount. Some do a hard dollar amount. Some do dollar amount or percentage matching based on years of service. Some do "as profits allow." Some do nothing.
You are correct that this money is only to the retirement account, it would not otherwise be cash in hand at a paycheck.
I'm not familiar with the UK, but in the US there is one important distinction between a pension and individual retirement accounts like a 401k, 403b, IRA, etc. Most pensions are defined benefits, meaning they are not controlled or accessible to the individual, they are, which are a promise for the future with zero individual control. The other instruments are ultimately owned and controlled by the individual.
You are typically limited to using the 401k your current employer offers if you want an employer contribution (which has a vesting schedule) and pre-tax investing, but when you change employers you can roll the money into another retirement instrument. The flip side is all responsibility for investment management, or even having retirement savings, is on the individual. Of course there are many people willing to help you manage it (for a fee).