That's usually referred to as a workplace pension or a group personal pension.
I have several pensions - I have a number of deferred workplace pensions, I have a deferred workplace final salary pension, I have one personal pension that is a stakeholder and one that is a SIPP. The latter are nothing to do with any employer and never have been. Nothing to do with auto enrolment.
If you're going to talk about financial products, you really need to get clear what you mean.
As someone who has set up pension and healthcare schemes for employers I am well aware of benefits packages. And the only time I've known an employer pay into a private pension is with very very high earners, for some very specific reasons, such as working abroad etc.
If the op has an employer with a workplace pension (which all workplaces do now) then she may have been auto enrolled and may get employer contributions (there are earnings limits so not everyone gets these) and the general advice is that everyone should take this up (with a few very odd exceptions) as it is "free"* money from your employer and could be advantageous for tax.
- It's not free really of course, you work for it, it's part of your remuneration.
But even then, you still have to decide what to invest it in, or just ignore it and let it sit in the default fund (which, to be fair, is what most people do).
Companies of a certain size have to offer a private pension
All companies with any number of employees have to set up a workplace pension. I run my own company with only me in it and even I have to in theory. I don't, but there are specific reasons for that.
Yes, the state pension age is going to rise. Also, the age you can draw from any pension is going to rise. This isn't some magic insightful prediction, the government has frequently said this!
But pensions are not anything clever, they are merely a wrapper with a certain tax treatment (a tax treatment subject to the whims of the government who just love fiddling with it and suddenly changing the retirement options of millions of people). You still need to decide how to invest the money in them.