Compression algorithm with multiple valid same-sized outputs

Is there a lossless compression algorithm that has hashing-like properties where there are multiple solutions to it?

As in for example, when a 1000-bit data-sequence is compressed into a 500-bit data sequence, there are multiple possible 500-bit data sequences that can be generated as outputs. Each of these 500-bit data sequences, once decompressed would all output the original 1000-bit data sequence. Additionally, upon decompression, any one of the 500-bit data sequences would also have multiple 1000-bit data sequence solutions upon decompression (around (2^1000/2^500) valid solutions). Collisions are fine, since otherwise it would be impossible for multiple 500-bit data sequences to be valid compression outputs of the 1000-bit data sequence.

The amount of "pigeon-holes" available for compression and decompression don't need to be symmetric. Also, it may not be appropriate to call a compression algorithm that produces multiple valid outputs when compressing and multiple valid same-sized outputs when decompressing as "lossless". But all the original data would still be there, it would just generate additional data on top of it. This additional data can be dealt with (the decompression would be stochastic if without help).

One more thing, the compression degree should be customizable (eg, from 1000-bits to 900-bits, or from 1000-bits to 950-bits, etc). Same with decompression. Otherwise, if such an algorithm requires the bit-size-change steps to be fixed, then it needs to be symmetric and fine enough to allow for multiple repeats. For ex, 1000-bit to 900-bit to 800-bit to 700-bit, etc. And the decompression process needs to backtrack for those exact sizes.

Is there a lossless compression algorithm that has hashing-like properties where there are multiple solutions to it?

No. A lossless compression algorithm† can't have two different inputs with the same output, contrary to a hash.

In the question, the proposition "Additionally, upon decompression, any one of the 500-bit data sequences would also have multiple 1000-bit data sequence solutions upon decompression" is wrong. In lossless compression, decompression (of the result of compression) is always deterministic. And "a 1000-bit data-sequence is compressed into a 500-bit data sequence" can apply to only at most $2^{500}$ out of the $2^{1000}$ existing 1000-bit data-sequences.

† That is, formally, a pair of algorithms called compression and decompression, each accepting finitely much data as input and producing finitely much data as output, such that feeding the output of compression to the input of decompression insures that the output of decompression exactly matches whatever the input of compression was.

The background idea in compression algorithms are due to the Typical Set and Asymptotic Equipartition Property (AEP) concepts from Information Theory.

So, roughly speaking, the compressed message is similar to a rewriting of its original symbols, like if some snippet, e.g. $0000:0000$ is more frequent in the original message, AEP gives us a measure of what is the most economical symbol that, uniquely, represent it, taking advantage on its redundant aspect: so, rewriting the same with fewer bits, and the general idea to reach that is building a mapping table, what is a completely deterministic thing.

So, that given reference we can read: "First, lets note that a lossless coding scheme needs to satisfy the following requirement:"

• Unique decodability: $x_i^n \neq x_i^m \implies C(x_i^n) \neq C(x_i^m)$