Tandem/twin circuit breakers installed where not allowed?
The existing tandems are of the correct model for this panel...
Your existing panel appears to be a very old GE panel, of a style that predates the current THQP half-width GE breaker, as well as the formal introduction of Circuit Total Limiting panels.  These ancient panels use type TR tandem breakers instead, and the spare breaker you photographed indeed is a type TR, in addition to looking identical from the face.
...but are indeed misplaced
This means that while the panel labeling only permits tandems in certain slots, neither the breakers nor the panel are keyed to enforce that, and thus you can get your current situation, where the installer slapped tandem breakers in wherever convenient.  This, while not an issue at the moment, can lead to problems with heat down the road if more tandems get shoved in there, stuffing the panel beyond its 20-circuit limit.  This is possible if 240V or multi-wire branch circuits are present, as those mean that the number of neutrals will not match the number of hots, rendering the 20-hole limit on the neutral bar irrelevant.
A millstone about your neck, if ye wish to keep it
Furthermore, the age of this panel means that it is one of the few cases where new production breakers of the labeled type are not compatible with the panel's busbar system. In other words, if you bought new-make GE THQL breakers for this panel, expecting them to fit and work, you'll be in for a major disappointment, as they likely won't even fit, as per this DIYChatroom post (pertinent parts quoted below for posterity):

GE changed the way they set the breaker's seating height. In my panel, you push the breaker down until the top of the molded slot rests on the stab. TQL breakers, THQL Issue UOxxx (probably MJxxx, NExxx and others too), and also Siemens QP and Eaton BR, can work this way, because they all have the same slot profile.
Recent panels have a ridge of insulating material next to the stab; the bottom of the breaker's case rests on the ridge. The stab is not as tall. It doesn't penetrate the breaker as deep. All the above breakers (except TQL which somehow doesn't quite fit) can be mounted. Also Issue RT-xxx of THQL, which doesn't fit the old bus because it can't accommodate the tall stab. Issue RT-xxx is all you will find in retail stock.

This means that your breaker selection is limited to NOS (new old stock) THQL breakers of Issues that will fit the panel, which is a dicey source of breakers at best, or to Eaton CL breakers that have been UL classified for use in GE panels (among others).  However, while the latter can be obtained reliably from any electrical supply house that deals in Eaton products, the UL classification process limits you to a small subset of possibilities.  Namely, using CL breakers limits you to a maximum of 50A branch circuits, and means you can't get double-stuff breakers, two-pole arc or ground-fault breakers, dual-function (arc fault/ground fault) breakers, or other exotica for your panel, either.
Nonetheless, if you wish to keep this panel, you can, provided it is in good condition on the inside still.  You'll have to rearrange this panel so that the double-stuff breakers are in slots designated for them, though, and replace that BR120 that currently sits in the bottom left with an Eaton CL120.  Note that you'll want to keep the spare TR breaker if you do this, too, as they are extremely difficult to find nowadays, since they have not been made in roughly half a century or so.
Replacement options
If you do decide to blow the money and replace this panel, you have a couple of options.  First, you could replace the whole panel with a 100A, 24-space or 30-space, main breaker panel.  This gives you a brand new panel, of a manufacturer of your choice, and plenty of room for future expansion as well.  Note that you'll have to replace the breakers wholesale to go with this panel swap, which may mean you have to put AFCIs in.
The other option would be to use a retrofit interior kit (mostly an Eaton thing, although Siemens now offers a retrofit interior as well) to replace the existing bus structure in the panel.  This would also require a wholesale breaker replacement, and requires costlier parts than replacing the panel outright, but has the advantage that the existing enclosure (cabinet) can be left in place.  This is useful if removing the old cabinet is impractical, something that is often the case for flush-mounted panels, or if rigid conduits come into the panel from both sides or the top and bottom.