Does Johnson have any similar ability?
Johnson has slightly more "hard power" than Mr. Trump, because Mr. Johnson is still a member of parliament (MP) while Mr. Trump held no elective office after he was succeeded by President Biden. But, Mr. Johnson has much less "soft power" than Mr. Trump does in the U.K. political system for reasons discussed below.
Can he support "pro-Johnson" candidates and oppose those who voted him
out, angling for another term as prime minister?
Boris Johnson left the main stage for the bleacher seats of a rank and file backbencher MP because he had lost the confidence of the British public and of his own political party elites. He hasn't completely lost support from rank and file members of his own party, perhaps, but he has lost all clout among the people who matter in his party who are considering his electability and the political consequences if he had continued to be P.M.
His net favorability with the general public in the U.K. dropped by about 82 points over about two years. Even the party rank and file started to turn on him in late 2021. The day before he resigned, 54% of 2019 Conservative Party voters stated in a survey that he should resign while only 33% took the position that he should stay, a net favorability of -21% which was about the same place he was in late 2021 with the party rank and file. (In contrast, Trump continued to be popular with both the rank and file and the elites in the Republican Party even after losing the 2020 Presidential election.)
On the eve of the intra-party vote for his replacement, however, about three weeks ago, Mr. Johnson was polling favorably compared to the candidates actually running for the Prime Minister position (a result that may have been, in part, a product of simple name recognition).
Mr. Johnson's resignation on July 7, 2022, was triggered by a wave of MPs resigning from his government in the face of the third major scandal his government had experienced in three years, which was the straw that finally broke the camels back. A month before he resigned:
Johnson survived a vote of confidence triggered by his own MPs amid
increasing dissatisfaction with his leadership. The secret ballot of
Tory lawmakers saw some 211 MPs vote in favor of Johnson, while 148
voted against him.
Basically, he left because his informal power was already gone or even negative among party elites (who are laser focused on winning the next election as opposed to merely their own or their own party's rank and file's policy preferences) and the general British public for a variety of reasons covered in other posts at Politics.SE (see also a list here citing three scandals in his government, a lack of focus and ideas, an unpopular tax increase, and inflation).
If Mr. Johnson tried to change his mind about stepping down from his position of power as P.M. voluntarily, he would be viewed as violating the norms of a member of a major British political party and would be seen as unacceptably disloyal to his party's leadership. If he did so, his party might not make him their candidate in his own riding going in the next election if he did not cease and desist from doing so upon request from the party leadership, and the party would probably deny him whatever perks of being an MP in his party that it could withdraw from him in the meantime.
Mr. Johnson has an obligation upon resigning and having a successor appointed under British political norms to get out of the way and allow his successor to do the job he freed up for her both as party leader and as PM for the U.K.
His only real power is to vote as he wishes in circumstances when his own party declared that a House of Commons vote is a "free conscience vote" which is not very often, or to join with other dissenting members of his own party in a "no confidence vote" en route to triggering a snap election if circumstance evolve negatively under the new PM and her cabinet.
While he can technically vote anyway he wants on a party line vote, the Conservatives have a sufficient majority that it wouldn't change the outcome, and since political parties in the U.K. don't have primary elections and instead choose who will run in each constituency in each election internally, regularly voted against the party line as a rank and file MP, would make it significantly less likely that Mr. Johnson would be allowed to run again as a Conservative MP again in the next election.
Also, since U.K. political parties don't hold primary elections, there is less room for Mr. Johnson to influence the selection of candidates to run in the next U.K. general election than there is in the U.S. where appeals to the rank and file members of the party can influence who is elected via a primary election.
Does his continued
presence in the House of Commons grant him a stage to do so?
If anything, continuing to be an M.P. and a member in good standing of his party imposes more obligations on what Mr. Johnson can do politically rather than giving him a platform from which he can have a megaphone to dissent.
If Mr. Johnson resigned from his M.P. seat and dropping his membership in the Conservative party, he would not longer have a normative expectation of loyalty to the party and could say what he wanted and build up a new band of loyalists under a new partisan banner or as an independent political voice.
Like France, which has an institution (the Constitutional Council) in which permanently retired former President's of the Republic are ex officio members in which they participate as "elder statesmen" the U.K. has "The Privy Council" to which members (including PM if for some reason they are not already one) are appointed for life. The Privy council carries out many day to day functions related to institutions, charities and companies who are incorporated by Royal Charter. It also is the formal mechanism by which parlimentary bills are presented for Royal Assent. Most of this work is carried out by active Parliamentarians (mainly members of the Government) of which Johnson remains one. Senior Members of the Privy Council form the Accession Council to formally appoint the new Monarch on the passing of the old one, as a former PM this will include Johnson.