Two ways to do this, in addition to what Andrew said. First, put your filter on one clip and tweak it until it looks good. Then, with that effect selected, go to Effects > Make Favorite Effect. Now, this filter with all its settings will be in your Favorites folder in the Effects window. Select all the clips on your timeline that you want to apply the filter to, then either drag the effect from your Favorites folder, or go to Effects > Favorite > Your Effect. This will apply it to all the clips you have selected.
Now, if you want the filter to auto-update on every clip without having to re-copy and re-paste it, the only way is to nest your edited sequence in a new sequence and apply the filter to that. However, this applies the filter to EVERY clip on every track. So, for instance, if you have video clips on Track 1 that should all have the filter applied, and some graphics on Track 2 that don’t need the filter, just nest everything in Track 1 in its own sequence. (Select everything on the track, then go to Sequence > Nest Item.)
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Exploring the pitfalls of the 5D workflow. The many, many pitfalls…