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As it is said in TFA, it did not have the structure of a computer, but it was a very complex digital automaton and it contained the typical digital circuits that are used today, like counters and shift registers, which at that time were a novelty.

The first digital circuits with vacuum tubes and gas tubes had been designed also in UK, during the decade preceding WWII, when most of them had been used in nuclear physics, e.g. in electronic counters for ionizing radiation, because the mechanical counters commonly used at that time were too slow for counting radiation events.

The earlier publications about such electronic counters were the inspiration both in UK for Flowers and the other people who contributed to the design of Colossus, and also in USA for the Atanasoff-Berry Computer, the first computer with vacuum tubes made in USA, whose electronic circuits were in turn an inspiration for the designers of ENIAC.

The first electronic computers were designed bottom-up, without much thinking about the high-level structure, until von Neumann handled that. The designers of those computers struggled mostly with the detailed schematic design of the digital blocks made with vacuum tubes, like flip-flops and gates, in order to make them work reliably, because before WWII there was very little experience with such circuits.

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It doesn’t resemble the computers of today, but none the less it was indeed a stored program digital computer. The first of its kind. The bombes were more akin to a special purpose calculation machine. Colossus was something a bit more special.
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‘Stored program’ is where the debate is. Colossus was ‘programmable’ but the program was not a series of instructions executed in a sequence - it was a single data process executed on the input data against a set of stored registers.

Definitely programmable, definitely capable of computing, but not a ‘stored program’ in the sense we know it today.

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Having seen the replica, I noticed it has a long continuous tape loop. Was the program "stored" in the paper tape?
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It was the first programmable electronic digital computing machine, yes. Special-purpose computing machines had been around since the 1600's (where they were used as textile weaving machines I think).
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And the “electronic” part is important. The Z3 predates in by 2 years and was a programmable digital computer, except it used electromechanical relays.

The Z3 is arguably more programmable than colossus and it was proven to be Turing complete in its stock configuration. Though the Z3 lacked the ability to conditionally branch, so nobody realised it at the time.

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