Can an 'over-stayed' stamp from one country affect my ability to travel to other countries?

Previously overstaying by less than 28 days will not attract any
mandatory refusal. However, although it should not affect any
application, it may be held against an applicant in terms of
credibility, depending on the Entry Clearance Officer.

Those who have previously overstayed by more than 28 days are
automatically refused entry clearance for different periods, depending
on whether or not they left voluntarily and at their own expense or at
the expense of the Secretary of State. The same periods apply to
those who entered the UK illegally and those who breached a condition
attached to their leave. Those who left the UK voluntarily, not at
the expense of the Secretary of State are barred only for one year.
Those who left the UK voluntarily, at the expense of the Secretary of
State are barred only for five years unless they left within six
months of being notified of the removal decision, in which case they
are barred for only two years. Those who were returned by the
Secretary of State involuntarily or deported are barred for ten years.

Where there has been more than one breach only the breach with the
longest exclusion period will be applied.

It should be noted that, in practice, the UKBA tends to serve notices
of removal on even those who are leaving the UK voluntarily. The
probable reason for this is so that the UKBA can record it as a
removal for statistical purposes. Any voluntary returnee should
therefore try to protect their position in advance by notifying the
UKBA of their intention to return and to keep evidence of this.

https://www.lawfirmuk.net/immigration_e_bad

While I agree with the first part of @MatthewHerbst’s answer, I strongly disagree with his suggestion to "lose" your passport, so take this as a very extended comment if you wish:

It depends a bit on your nationality whether you need a Schengen / UK visa or are eligible for a visa / leave to remain on arrival.

However in both cases either in the visa application and/or at immigration you might be asked whether you had previously overstayed or have or had a ban from some country. It is a very bad idea to lie in this context.

While "losing" a passport might initially sound like a smart idea especially in the case that you do not need to apply for a visa, I highly recommend against it. Losing your passport means there will be a police report/flag somewhere about a lost passport and security agencies are sensitive to this, as a similar strategy is also used by "terror tourists" to camps in Afghanistan/Pakistan/Syria or wherever else the world is on fire right now. There is a chance that immigration is getting wind of this and will give you extra scrutiny. At this point I also would not at all be surprised if the authorities of South Africa and the UK but also Schengen shared access to their databases and already knew about your overstay and now will be very curious as to why you "lost" your passport so recently.

Let me conclude by citing from this excellent answer on another question, citing UK procedures:

The strategy of concealing an adverse immigration event by ‘losing’ one’s passport and getting a fresh, unblemished passport is a poor one. There is a history associated with the passport that is not accounted for in its physical pages, but rather in computer systems linked to the passport number. And a new passport will contain a record linking it back to the previous one. People still try this strategy, however, and when they get caught the results are catastrophic.

If the UK catches somebody doing this, they will get logged (along with their biometrics) as violating Paragraph 320 of the Immigration Rules. That usually means the person can forget about coming here for a long time, if ever. Plus they will tell the US Department of Homeland Security about it (regardless of your nationality). They keep your biometrics on file for 12 years if you have a clean history, and indefinitely if you fall under Paragraph 320.

Yes, the immigration officer can see all the stamps in your passport, and can, at their discretion, refuse you entry into the country (even if you have a valid visa to enter). Further, some countries share databases with this information that might effect your ability to get a visa in the first place.

If you just want to get rid of the overstay stamp, you should “lose” your passport and get a new one. You won’t be able to remove your information from a database though if it is in one, or restrict who that information is shared with.