How to handle independent variable in regression where no relationship exists between samples I would like to use a multinomial logistic regression to get win probabilities for each of the 5 horses that participate in any given race using each horses previous average speed.
RACE_ID    H1_SPEED     H2_SPEED    H3_SPEED    H4_SPEED    H5_SPEED    WINNING_HORSE
1          40.482081    44.199627   42.034929   39.004813   43.830139   5
2          39.482081    42.199627   41.034929   41.004813   40.830139   4

I am stuck on how to handle the independent variables for each horse given that any of the 5 horses average speed can be placed in any of H1_SPEED through H5_SPEED.
Given the fact that for each race I can put any of the 5 horses under H1_SPEED meaning there is no real relationship between H1_SPEED from RACE_ID 1 and H1_SPEED from RACE_ID 2 other than the arbitrary position I selected.
Would there be any difference if the dataset looked like this -

*

*For RACE_ID 1 I swapped H3_SPEED and H5_SPEED and changed WINNING_HORSE from 5 to 3

*For RACE_ID 2 I swapped H4_SPEED and H1_SPEED and changed WINNING_HORSE from 4 to 1
RACE_ID    H1_SPEED     H2_SPEED    H3_SPEED    H4_SPEED    H5_SPEED    WINNING_HORSE
1          40.482081    44.199627   43.830139   39.004813   42.034929   3
2          41.004813    42.199627   41.034929   39.482081   40.830139   1

Is this an issue, if so how should this be handled? What if I wanted to add more independent features per horse?

If it is not possible to link the arbitrary horse numbering by race to a literal horse through some lookup table (which it sounds like it is not), it will not be possible to get meaningful probabilities for the horse numbers as they do not refer to the same horse across races.
It sounds like, across races, there may be multiple features for each horse other than speed.  If these 5 horse numbers always refer to the same literal horses (with arbitrary labels), and there are multiple features per race, it may be worthwhile to attempt to use those all the features to match the arbitrary horse labels to across races to harmonize labels for a literal horse.
The idea would then be for each race to attempt to match the horse labels based on the values of all the features.  An approach like this would make the strong assumption that the horses have a similar profile across races in terms of the values collected on each horse in each race.  That said, it may be the only way to get the data in shape for subsequent modeling.
In effect, what I'm recommending is a kind of fuzzy matching/data linking approach (see probabilistic data linking section at the wiki here).
Regarding the second question, changing the labels should not matter given they are (currently) arbitrary. Again, the important thing to do to make the probability estimates meaningful is to harmonize the literal horse to horse number mapping across races.
