TLDR: It's not indicating bad management, but it might mean they are not a good fit for you.
It's not a bad company. And Skype is not considered spyware.
Sadly a decent amount of user tracking and convenience over anonymity / user liberty/ security / data protection is the norm these days.
They have their company solution to do interview (and other calls) and they follow that. I'd find it a bit inflexible to not fall back to a normal telephone conversation, but that they are not too keen on installing other software is totally normal. Some companies where individual employees have a lot of individual responsibility and freedom might accept that, but I would not expect that you can dictate to any company how the interview is done.
Now what does it say about the company? It's probably decently large. The larger a company gets the more it tends to follow policies / company wide solutions to standard problems and adhere to them. There are exceptions where individuals / individual departments have large freedoms, but this seems to be a company that has a tight organisation - at least if the HR department is indicative of the rest of the company.
So while it's not ultimately bad, it might just not be an environment where you would want to work. If you prefer flexibility, individual decisions and or a focus on data protection / user empowerment / open source software, then this could indicate that this is not the right company for you. However, sometimes different departments in the same company are run totally different. While the HR department might have strict rules and use mass market software, other departments might be run different. E.g. in many companies the default operating system is Windows (along with central administration), but people in certain departments are allowed to install their own OS on their work machine and manage their machine themselves (but don't get the same support in maintaining it). Whether this is the case, you won't find out if you don't get an interview.
But if you have enough other options, then this can be a quick filter that sorts out companies you likely don't fit with anyway.
Options to manage the problem as good as possible: As many companies do use Skype for their remote interview calls, you might want to try and get past that step to see if they are generally a bad fit or whether this is the only thing you don't like while minimising the "impact" on you. I'm assuming here that you either don't want to install Skype on your machine, as you consider it spyware, and/or that you don't want to identify yourself towards Microsoft when using Skype.
- Ask to use normal telephone instead (in this case you tried and did not succeed, but still this, is relatively likely an option,at least compared to suggesting other software based options)
- Use web-skype, that way you avoid an installation of something you consider spyware, you still require an account though
- If you feel really paranoid about it, you might also use a Linux distribution on a USB stick to boot into that instead of your normal OS and either install Skype there or use web skype from that system
- if you want to avoid providing your real phone number when creating a throwaway account, you can try to use an online service that allows you to read sms to mobile numbers provided by them online (google for concrete services); however, those numbers are often black-listed by big companies soon after they go online it seems. So you might have to try a few times till you manage to open an account with one of them. And you run the risk that it is shut-down by Skype should they find out afterwards (no idea what their policy is regarding the usage of such phone numbers). You might want to make sure that is not illegal where you live depending on how harsh your government looks at attempts to stay anonymous.
- buy a cheap/used phone with a prepaid sim card, install Skype there and use it only for your interviews, then get rid of both (or re-install the system on the phone), same option applies to a laptop
- ask in an internet cafe if they can install Skype for you; even if they don't have it on there as a default, they might be fine with installing it.
Personal Opinion: To me it's funny that Skype is now adapted all over the place in corporations when it stopped being the leading solution that works without problems cross-platform quite a while ago. Due to that one might see a fixation on Skype as a (tiny) indicator of a company trying to catch up with the times, but still quite a bit behind. But I personally wouldn't use that as a sole discriminator to go ahead or not with an interview process.