Yes, cloning, repeatedly, maintained, could make you immortal.
There's some logistical concerns that make this trickier than the spell itself:
Vessel must be undisturbed
So, ideally, you set up a nice young version of yourself, hide it away for the time something goes wrong and go about your life, right? Well, the longer it's around, the more likely, over time, something COULD happen to it. Especially in a world where you've got things like purple worms, umber hulks and bulettes and other critters that dig through granite like butter.
Well, then it makes sense to set up some defensive measures, right? Traps, spells, etc. But in the world of D&D, the more defensive measures you put up, the more people assume it's got something valuable to steal...
Now, as a GM I wouldn't just automatically assume something is going to happen, but if the clone is sitting around for decades, or the wizard in question has enemies seeking them out, then we'd start having to think about problems.
A giant diamond
So, the diamond is worth 1,000 gp. This doesn't mean you can simply pull out 1,000 gp and find these diamonds anywhere, everywhere. There's got to be a limited number of them. You're probably not the only caster who is looking into this spell.
So, a bunch of wizards want immortality from a limited resource and are all looking for it.
I'm sure that's not going to lead to problems.
Welcome to the Immortal Club
So, if you manage to live far beyond even what most D&D world folks know people to be capable of, and you're known to be an awesome wizard... how many other people are going to be trying to get your secret of immortality from you?
How do other things which are immortal feel about this? Do they find a way to manipulate/play you because you're new to this game? Do they already have a control on the 1,000 gp diamonds and dole them out to the few wizards who have Clone just to keep them under their leash?
Is there an alliance of lichs who are jealous you've found a way to live, but actually live, not undead live, and they'd like to simply stomp you down for being audacious?
Are there mind flayers looking to eat the juicy mind of a super-intelligent wizard with 800 years of tasty-tasty knowledge?
Are there divine guardians of life and death who did their accounting and finding there's a soul short that needs to move on?
Gameplay
Unless you're playing a very unusual game of D&D, these issues aren't likely to come up too much simply because the timescale is too short. But it makes excellent source of adventures based on NPCs - just imagine what happens when you do have a wizard who has been doing this and dealing with all of these problems and what that means for the PCs when they get involved in it.
Recall that the clone is inert before it's activated — it has no personality, memories, or abilities of any kind at that point. The paragraph about the clone's creation doesn't mention memories yet; that comes later in the spell's process. So it's empty and inert, and isn't the same as the original yet.
The part of the spell description that does grant the clone personality, memories, and abilities is written immediately after it describes the soul of the original animating it — it's in this paragraph about death and animation that the spell first mentions that the original and the clone have the same memories. The construction of the paragraph indicates a causal link: that they match because (and therefore when) the soul has animated the backup body.
So no, you don't lose memories when reincarnating in your clone.
Best Answer
Looks like it can work, RAW
The duration of Clone is listed as "Instantaneous", which leans us toward saying No, Soul Cage can't work on someone with a Clone prepared. However, Soul Cage is a reaction you take to a predetermined set of circumstances that you have prepared for--not unlike Counterspell. The triggers for Soul Cage may be harder to spot (final moment of death as opposed to a person making the motions of casting a spell), but an observant and prepared caster might still catch it.
Since Counterspell can counter other spells with a duration of Instantaneous, I would say the similarly reaction-cast Soul Cage would work the same way, in that the caster noticed a particular trigger and got the spell off at the last split second and Soul Cage grabs the soul before the transfer can complete.
Assuming Soul Cage does work, the soul would finish the transfer to the clone as soon as Soul Cage wears off.
I went and looked up the spells in question (PHB p. 222, XGE p. 165). Since Spells Do What They Say They Do, the Clone spell sets up four prerequisites for the soul to transfer to the prepared clone: the clone is mature, the original body is dead, the soul is free, and the soul is willing. There is no time limit given between "original body is dead" and the soul transferring for the spell to work successfully, and the existence of the "soul is free" stipulation means there are cases where a soul could be trapped upon death instead of just automatically transferring.
Also, note that Soul Cage, as written, just releases the soul when you are done with it, no text about using up the soul or where it's released to.
So, if you've successfully snatched a soul via Soul Cage and the spell has since ended, the soul is just released freely. Now, all four conditions of the Clone spell are true, and the soul can complete the transfer.
Still talk to your DM about it to see if you can do it. Not all DMs go by RAW all the time, and in the process of writing this answer I came up with three or four ways a DM ruling or houserule would totally alter the interaction of spells and soul here.