Is this kind of settlement agreement a redundancy?
No. Formally speaking, ending employment with a settlement agreement is not a redundancy. In this context the settlement agreement is a contract between employer and employee to end the employment, the employee waives the right to pursue any legal action against the employer and the employer pays the employee some money to compensate for the ending of employment and the waiving of rights.
A redundancy is when the role - not the employee - is redundant because of some change in the organisation's circumstances (e.g. not enough work for the role, changing location, insolvency or closure).
Is the employer allowed to to this?
Generally speaking, yes they are.
Note that, from the outside, this looks like an agreement between two consenting parties - generally speaking, people/companies can agree things between themselves.
Why might an employer offer a settlement agreement instead of redundancy?
Some employers use settlement agreements to terminate employment because (a) this avoids the best practises and obligations of the formal redundancy process and (b) mitigates the risk of claims of unfair dismissal.
Redundancy might require more effort than they'd prefer, particularly if they want to dismiss 20 or more employees within 90 days (when they would have to do rounds of consultations).
Compensation
In terms of a settlement agreement the employee is free to ask for more than what is offered - they might not get more but they are free to ask.
Redundancy pay depends on a formula that will output one result for a given set of circumstances, whereas a settlement agreement can be negotiated until it is signed.
A settlement agreement might offer more than what could be gained by redundancy (particularly statutory redundancy, i.e. the legally mandated minimum) to incentivise the employee to take it instead of pursuing some other avenue. This is because the employer perceives the settlement route to be better for the employer than the redundancy route.
An employee offered a settlement agreement should compare its compensation with what they are entitled to by way of redundancy (whether contractual or statutory) and other practical or feasible outcomes, and weigh up their options.
The employee might also consider asking for the agreement to include non-monetary things such as "the employer will provide a good reference".
Note that a person employed for less than two years is not entitled to statutory redundancy pay and they don't have the same rights to claim unfair dismissal as a person employed for more than two years.
Where can someone get advice specific to their circumstances?
The employer should make sure the employee has received independent legal advice and understands the agreement before signing the agreement. Sometimes employers pay for or contribute to the cost of the independent legal advice.
In the UK, you can contact ACAS for advice, costing no more than the cost of the phone call.
Some solicitors offer a free initial consultation, perhaps taking up to 30 minutes - you tell them the circumstances and they outline your options. It is up to you what you do next.